Skip to main content

Full text of "The London, Edinburgh and Dublin philosophical magazine and journal of science"

See other formats


554 

I Sc ienti fic Library % 

— I 




OMIIEB STATES PATENT OFFICE 



11 — 8620 



government frintixq office 



THE 
LONDON, EDINBURGH, and DUBLIN 

PHILOSOPHICAL MAGAZINE 

AND 

JOURNAL OF SCIENCE. 



CONDUCTED BY 

SIR OLIVER JOSEPH LODGE, D.Sc, LL.D., F.R.B. 

SIR JOSEPH JOHN THOMSON, O.M., M.A., So.D., LL.D., F.R.S. 

JOHN JOLY, M.A., D.Sc, F.R.S., F.G.S. 

RICHARD TAUNTON FRANCIS 

AND 

WILLIAM FRANCIS, F.L.S. 



"Nee araisearuni sane textus ideo melior quia ex se ftla gigiiunt, nee noster 
ilior quia ^x alienis libamus ut apes." Just. Lips. Polit. lib. L cap. 1. Not. 



VOL. XLIV.— SIXTH SERIES. 
JULY— DECEMBER 1922. 



LONDON-. 
TAYLOR AND FRANCIS, RED LION COURT. FLEET STREET. 

SOLD BY SMITH AND SON, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN ; - 
AND VEUVE J. BOYVEAU, PAKI C , 






"Ateditatiuins est perscrutari occulta ; contemplationis est admirari 
perspicua '. . . . Admiratio generat qiuestionem, quzestio investigationero, 
investigalio inventionem." — Hugo de S. Victore. 



— — " Cur spirent venti, cur terra dehiscat, 
Our in are turgescat, pelago cur tantus amaror, 
Cur caput obscura Phoebus ferrugine condat, 
Quid toties diros cogat -flag-rare cometas, 
Quid pariat nubes, veniant cur fulmina coelo, 
Quo micet igne Iris, superos quis conciat orbes 
Tain vario motu." 

J, B. Pinelli ad Muzonium. 



ATiERK 




CONTENTS OF VOL. XLIV 

(SIXTH SERIES). 

'■221. 



NUMBER CCLIX.— JULY 1922. 

Mr. A. R. McLeod on the Unsteady Motion produced in a Uniformly 
Rotating Cylinder of Water by a Sudden Change in the Angular 

Velocity of the Boundary 1 

Dr. Norman Campbell on the Elements of Geometry 15 

Dr. Dorothy Written on the Rotation of Slightly Elastic Bodies ... 30 
Mr. G. H. Henderson on the Straggling of ex. Particles by Matter . . 42 

Dr. G. Green on Fluid Motion relative to a Rotating Earth 52 

Prof. G. N. Antonoffon the Breaking Stress of Crystals of Rock-Salt. 62 

Dr. Norman Campbell on the Measurement of Chance 07 

Messrs. H. T. Tizard and D. R. Pye : Experiments on the Ignition 

of Gases by Sudden Compression. (Plate I ) 79 

Mr. C. Rodgers on the Vibration and Critical Speeds of Rotors. . . . 122 
Mr. P. Cormack on Harmonic Analysis of Motion transmitted by 

Hooke's Joint 156 

Messrs. E. W. B. Gill and J. H. Morrell on Short Electric Waves 

obtained by Valves 161 

Sir George Greenhill on Pseudo-Regular Precession 179 

Dr. J. \Y. Nicholson on the Binding of Atoms by Electrons 193 

Mr. L. St. C. Broughall on Theoretical Aspects of the Neon Spectrum. 2U4 
Dr. F. H. Newman on Absorption of Hydrogen by Elements in the 

Electric Discharge Tube 215 

Mr. B. A. M. Cavanagh on Molecular Thermodynamics. II 226 

Prof. J. G. Gray on the Calculation of Centroids 247 

Messrs. A. P. II. Trivelli and L. Righter : Preliminary Investiga- 
tions on Silberstein's Quantum Theory of Photographic Exposure. 252 
Dr. L. Silberstein on a Quantum Theory of Photographic Exposure. 257 
Mr. R. F. Gwyther on an Analytical Discrimination of Elastic 

Stresses in an Isotropic Body 274 

Mr. II. S. Rowell on Damped Vibrations 284 

Notices respecting New Books : — 

Mr. L. F. Richardson's Weather Prediction by Numerical 

Process 285 

Proceedings of the Geological Society : — 

Mr. C. E. No will Bromehead on the Influence of Geology on 

the History of London 286 

Intelligence and Miscellaneous Articles-. — 

On Young's Modulus and Poisson's Ratio for Spruce, by Mr. II. 
Carrington 288 



IV CONTENTS OF VOL. XL1V. SIXTH SERIES. 

NUMBER OOLX.— AUGUST. 

Page 

Mr. C. J. Smith on the Viscosity and Molecular Dimensions of 
Gaseous Carbon Oxy sulphide (COS) 289 

Prof. A. 0. Rankine on the Molecular Structure of Uarbon Oxy- 
sulphide and Carbon Bisulphide 292 

Mr. F. P. Slater on the Rise of y Ray Activity of Radium Emanation. 300 

Profs. J. N. Mukherjee and B. C. Papaconstantinou on an Experi- 
mental Test of Smoluchowski's Theory of the Kinetics of the 
Process of Coagulation 305 

Prof. J. N. Mukherjee on the Adsorption of Ions 32L 

Prof. W. M. Hicks on certain Assumptions in the Quantum-Orbit 
Theory of Spectra 346 

Mr. F. C. Toy on the Theory of the Characteristic Curve of a Photo- 
graphic Emulsion 352 

Mr. A. M. Mosharrafa on the Stark Effect for Strong Electric 
Fields ■ , . 371 

Mr. E. Takagishi on the Damping Coefficients of the Oscillations in 
Three-Coupled Electric Circuits 373 

Prof. S. C. Kar on the Electrodjnamic Potentials of Moving 
Charges ■ . . . 376 

Mr. A. E. Harward on the Identical Relations in Einstein's Theory. 380 

Mr. H. S. Rowell un Energy Partition in the Double Pendulum . . . 382 

Piof. J. S. Townsend on the Velocity of Elections in Gases 384 

Prof. H. A. McTaggart on the Electrification at the Boundary 
between a Liquid and a Gas 386 

Prof. L. V. King on a Lecture-Room Demonstration of Atomic 
Models. .Plate II.) 395 

Mr. 11. D. Murray on the Influence of the Size of Colloid Particles 
upon the Adsorption of Electrolytes 401 

Notices respecting New Books: — 

Dr. G. Scott Robertson's Basic Slags and Rock Phosphates . . . 415 



NUMBER CCLXI.— SEPTEMBER, 

Prcf. Sir E. Rutherford and Dr. J. Chadwick on the Disintegration 
of Elements by oc Particles 417 

Prof. W. L. Bragg and Messrs. R. W. James and 0. II. Bosanquet 
on the Distribution of Electrons around the Nucleus in the Sodium 
and Chlorine Atoms 433 

Messrs. C. G. Darwin and R. H. Fowler on the Partition of 
Energy 450 

Mr. M. II. Belz on the Heterodyne Beat Method and some Appli- 
cations to Physical Measurements 479 

Mr. R. F. Gwyther on the Conditions for Elastic Equilibrium under 
Surface Tractions in a Uniformly Eolotropic Body 501 

Mr. C. J. Smith on the Viscosity and Molecular Dimensions of 
Sulphur Dioxide 508 

Mr. S. Lees on a Simple Model to Illustrate Elaslic Hysteresis. . . . 511 

Prof. R. W. Wood on Atomic Hydrogen and the Buhner Series 
Spectrum 538 

Mr. D. Coster on the Spectra of X-rays and the Theory of Atomic 

• Structure 546 

Pro£ E. H. Barton and Dr. H. M. Browning on Vibrational Re- 
' sponders under Compound Forcing. (Plates III. & IV.) ...... 573 



CONTENTS OF VOL. XLIV. SIXTH SERIES. V 

Page 

Dr. Norman Campbell and Mr. 13. P. Dudding on the Measurement 
of Light 577 

Mr. D. L. llammick on Latent Heats of Vaporization and 
Expansion 590 

The late William Gordon Brown on the Faraday-Tube Theory of 
Electro-Magnetism , . 594 

Mr. B. A. M. Cavanagh ou Molecular Thermodynamics. Ill 610 

Prof. A. W. Porter and Mr. J. J. Hedges on the Law of Distribution 
of Particles in Colloidal Suspensions, with Special Reference to 
Perrin's Investigations , (541 

Mr. St. Landau-Ziemecki on the Emission Spectrum of Monatomic 
Iodine Vapour 651 



NUMBER COLXII.— OCTOBER. 

Sir J. J. Tuomson : Further Studies on the Electron Theory of 

Solids, The Compressibilities ot a Divalent Metal and of the 

Diamond. Electric and Thermal Conductivities of Metals 657 

Dr. G. H. Henderson on the Decrease of Energy of ol Particles on 

passing through Matter 680 

Mr. D. C. Henry on a Kinetic Theory of Adsorption 689 

Prof. S. R. Milner on Electromagnetic Lines and Tubes 705 

Mr. A. Bramley on Radiation 720 

Mr. G. Breit on the Effective Capacity of a Pancake Coil 729 

Prof. F. E. TIackett on the Relativity-Contraction in a Rotating 

Shaft moving with Uniform Speed along its Axis 740 

Dr. T. J. Baker on Breath Figures 752 

-Mr. A. Sellerio on the Repulsive Effect upon the Poles of the 

Electric Arc 765 

Mr. B. B. Baker on the Path of an Electron in the Neighbourhood 

of an Atom 777 

Prof. A. W. Porter and Mr. R. E. Gibbs on the Theory of Freezing 

Mixtures , . 787 



NUMBER CCLXIII.— NOVEMBER. 

Mr. G. Shearer on the Emission of Elections by X-Rays 793 

Mr. A. J. Saxton on Impact Ionization by Low-Speed Positive 

H-Ions in Hydrogen , 809 

Messrs. C. G. Darwin and R. i\. Fowler on the Partition of Energy. 

— Part II. Statistical Principles and Thermodynamics 823 

Mr. J. H. Van Vleck on the normal Helium Atom and its relation 

to the Quantum Theory 842 

Mr. G. A. Tomlinson on the Use of a Triode Valve in registering 

Electrical Contacts 870 

.Mr. E. A. Milne on Radiative Equilibrium : the Insolation of an 

Atmosphere 872 

Dr. S. C. Bradford on the Molecular Theory of Solution. II 897 

Mr. R. A. Mallet on the Failure of the Reciprocitv Law in 

Photography 904 

Me-srs. R. W. Rolens, J. II. Smith, and S. S. Richardson on 

Magnetic Rotatory Dispei s'on of certain Paramagnetic Solutions. 912 



VI CONTENTS OF VOL. XLIV. SIXTH SERIES. 

Page 
Dr. F. W. Edridge-Green on Colour- Vision Theories in Relation 

to Colour-Blindness 916 

Mr. A. H. Davis on Natural Convective Cooling in Fluids 920 

Mr. A. II. Davis on the Cooling Power of a Stream of Viscous 

Fluid : 940 

Dr. F. H. Newman on a Sodium-Potassium Vapour Arc Lamp. 

(Plate V.) 944 

Mr. J. J. Mauley on the Protection of Brass Weights 948 

Mr. H. S. Rowell on the Analysis of Damped Vibrations 951 

Mr. F. M. Lidstone on the Full Effect of the Variable Head in 

Viscosity Determinations , 953 

Dr. L. Silberstein and Mr. A. P. II. Trivelli on a Quantum Theory 

of Photographic Exposure. (Second Paper.) ... 956 

Dr. J. S. G. Thomas on the Discharge of Air through Small Orifices, 

and the Entrainment of Air by the .Issuing Jet. (Plate VI.). . . 969 
Dr. J. R. Partington on the Chemical Constants of some Diatomic 

Gases 988 

Mr. M. F. Skinner on the Motion of Electrons in Carbon Dioxide. . 994 
Mr. W. N. Bond on a Wide Angle Lens for Cloud Recording. 

(Plate VII.) . . . 999 

The Research Staff of the General Electric Company Ltd., London : 
A Problem in Viscosity : The thickne-s of liquid films formed on 

solid surfaces under dynamic conditions. (Plate VIII.) 1002 

Prof. S. Timoshenko on the Distribution of Stresses in a Circular 

Ring compressed by Two Forces acting along a Diameter .... 1014 

Prof. A. VV. Porter on a Revised Equation of State 1020 

Prof. V. Ka rape toff : General liquations of a Balanced Alternating- 
Current Bridge 1024 

Prof. J. S. Towtjsend and Mr. V. A. Bailey on the Motion of Elec- 
trons in Argon and in Hydrogen 1033 

Prof. S. R. Milner : Does an Accelerated Electron necessarily radiate 

Energy on the Classical Theory ? .\ ..... . 1052. 

Mr. A. Press on a Simple Model to illustrate Elastic Hysteresis . . 1053 

Mr. S, Lees : Note on the above = 1054 

Notices respecting New Books : — 

M. H. Ollivier's Cours de physique generale 1055 

Dr. J. W. Mellor's A Comprehensive Treatise on Inorganic 

and Theoretical Chemistry - 1056 

Mr. S. G. Starling's Science in the Service of Man : Electricity. 1058 
M. Gustave Mie's La theorie Einsteinienne de la Gravitation. . J 058 
Wave-lengths in the Arc Spectra of Yttrium, Lanthanum, and 
Cerium and the preparation of pure Rare Earth Elements. . 1058 

The Journal of Scientific Instruments 1059 

Proceedings of the Geological Society : — 

Mr. R. D. Oldham on the Cause and Character of Earthquakes. 1060 

Sir C. J. Holmes on Leonardo da Vinci as a Geologist 1061 

Intelligence and Miscellaneous Articles: — 

On the Buckling of Deep Beams, by Dr. J. Prescott 1062 

On Damped Vibrations, by Mr. C. E. Wright 1063 

On the Magnetic Properties of the Hydrogen - Palladium 

System, by Mr. A. E. Oxley 1063 

On Short Electric Waves obtained by Valves, by Prof. R. 
Whiddington 1064 



CONTENTS OF VOL. XLIV.— SIXTH SERIES. VII 

NUMBER CCLXIV,— DECEMBER. 

Page 

Mr. 1!. Hargreaves on Atomic Systems based on Free Electrons, 
positive and negative, and their Stability 1065 

Prof. R. AY. Wood on Selective Reflexion of X. 2536 by Mercury 
Vapour 1105 

Prof. R. W. Wood on Polarized Resonance Radiation of Mercury 
Vapour. (Plate IX.) . ._ _ 1107 

Mr. S. J. Barnett on Electric Fields due to the Motion of Constant 
Electromagnetic Systems 1112 

Prof. Megh Nad Saba on the Temperature Ionization of Elements 
of the Higher Groups in the Periodic Classification 1128 

Prof. F. Uorton and Dr. A. C. Davies on the Ionization of Ab- 
normal Helium Atoms by Low-Voltage Electronic Bombardment. 1140 

Prof. J. S. Townsend on the Ionizing Potential of Positive Ions . . 1147 

Dr. G. Breit on the Propagation of a Fan-shaped Group of Waves 
in a Dispersing Medium 1149 

Prof. E. Keighlley Rideal on the Flow of Liquids under Capillary 
Pressure '.1152 

Prof. S. Russ and Mr. L. H. Clark on a Balance Method of mea- 
suring X-Rays 1 1 59 

Mr. J. W. T. Walsh on the Measurement of Light 1165 

Notices respecting New Books : — 

The Cambridge Colloquium, 1916 : Part 1 1169 

Mr. J. Edwards's A Treatise on the Integral Calculus : Vol. II. 1169 

Index 1170 



PLATES. 

I. Illustrative of Messrs. H. T. lizard and D. R. Pye's Paper 0:1 
Experiments on the Ignition of Gases by Sudden Com- 
pression. 
II. Illustrative of Prof. L. V. King's Paper on a Lecture-Room 
Demonstration of Atomic Models. 
III. & IV. Illustrative of Prof. Barton and Dr. B:owning's Paper on 
Vibrational Responders under Compound Forcing. 
V. Illustrative of Dr. F. H. Newman's Paper on a Sodium- 
Potassium Vapour Arc Lamp. 
VI. Illustrative of Dr. J. S. G. Thomas's Paper on the Discharge 
of Air through Small Orifices, and the Entrainment of Air 
by the Issuing Jet. 
VII. Illustrative of Kir. W. N. Bond's Paper on a Wide Angle 

Lens for Cloud Recording. 
VIII. Illustrative of The Research Staff of the General Electric 
Company on a Problem in Viscosity. 
IX. Illustrative of Prof. R. W. Wood's Paper on Polarized Reso- 
nance Radiation of Mercury Vapour, 



THE 
LONDON, EDINBURGH, and DUBLIN 

PHILOSOPHICAL MAGAZINE 

AND 

JOURNAL OF SCIENCE. 




I. The Unsteady Motion 'produced in a Uniformly Rotating 
Cylinder of Water by a Sudden Change in the Angular 
Velocity of the Boundary. By A. R. McLeod, M.A., 
Fellow of Gonville and Caius College , Cambridge *. 

IN the following paper a comparison is made between 
the observed angular velocities in a rotating circular 
cylinder of water, and those calculated by two-dimensional 
theory in which the effects of the base and the free surface 
are neglected, and each particle of water is assumed to move 
in a circle about the axis of rotation. The two cases dealt 
with are those of unsteady motion in which the cylindrical 
containing-wall is suddenly started from rest, or suddenly 
stopped when, with the water, it is rotating uniformly. The 
measurements are all of surface velocities, because the use of 
lycopodium particles floating on the surface was necessary, 
as liquid globules of the same density as the water would 
not remain at the same depth for any length of time. 
Three cylinders of approximate diameters 5 cm., 15 cm., 
and 25 cm. were used, and three angular velocities, viz. 
36 r.p.m., 10 r.p.m., and 1^ r.p.m. The small cylinder 
was four diameters, and the other two cylinders were two 
diameters long. The observed velocities show a departure 
from theory, which increases with angular velocity and 
with the size of the cylinder, but which tends to vanish 
at very low speeds. The discrepancy is much greater 

* Communicated by the Author. 
Phil Mag. S. 6. Vol. 44. No. 259. July 1922. B 



2 Mr. A. R. McLeod on Unsteady Motion produced 

for the " stopping " than for the " starting " experiments. 
The work was done at the Royal Aircraft Establishment, 
Farnborough, during the months Feb.-Sept. 1919. 

In some later work, not yet published, the discrepancy 
in the case of the " starting " experiments is traced to 
the effect of the base ; and the theory will probably apply 
to this motion in very long cylinders, provided eddies 
do not form owing to initial instability of the water. In 
the case of the " stopping " experiments, the discrepancy 
is due to instability and turbulence. 

§ 1. Theoretical. 

In dealing with a rectilinear two-dimensional eddy in an 
incompressible fluid which contains no sources or sinks, 
the usual assumption is that particles of the fluid move 
in circles about the axis of the eddy. This makes the 
problem one of complete symmetry, and the radius vector r 
and the time t are the two independent co-ordinates. The 
equations of motion, when written in cylindrical co-ordinates 
with the axis of z along the axis of symmetry, reduce to tlie 
following forms, in which p denotes pressure, p is density, 
v is the kinetic viscosity, and <fi is the angular velocity 
about the axis : — 



^ = p^' 2 (i) 



for the pressure, and 



dr* r "dr v ~dt 



(2) 



giving the angular velocity. Let us suppose that the 
angular velocity <j> satisfies the conditions 

= F(r) for t= 0, . . (3) 

</> = <f)(t) „ r — c = radius of cylinder ; . (4) 

that is, at the initial instant the angular velocity in the 
cylinder is known to be F(V) at radius r, while thereafter 
the rotation of the boundary is prescribed to be <j>(t). The 
solution of (2) satisfying the conditions (3) and (4) is, in 






by Change in Angular Velocity. 3 

terms of a series of Bessel functions : 

«=i ' i'' o v a «y / Jo 

-j W,(«W«)«-"^ fWy,.^, . (5) 



a=] 



rJ (a„) 



f ${r)e va n* T l*dT, 



where J 1 (a )l )=0 and u n is the nth root of this Bessel 
function of the first order. 

Taking </>(£) =Ii = constant and F(r) = 0, we have 



O, r n=1 ««J 2 («yi) 



(6) 



This is the solution for the case in which the water is 
initially at rest and the cylinder suddenly rotates with a 
constant angular velocity fl. The solution is given as 
a problem in Gray & Mathew's ' Treatise on the Bessel 
Functions ' (Ex. 38, p. 236, 1st edn.). 

Taking </>(£) = and F(r) = XI = constant, we find the 
solution 

(j> _2c S Ji(* n r/c) H , C 2 

12 ?< w= i u n 2 {tx n ) 

This is the solution for the case in which the water is 
initially rotating with a constant angular velocity O, and 
the cylinder is suddenly stopped. The solution is given by 
Stearne, Q.J. Math. xvii. p.' 90 (1881), and Tumlirz, Sitz. d. 
k. Alcad. in Wien, lxxxv. (ii.) p. 105 (1882). 

The phenomena which (6) and (7) are supposed to 
represent are at the basis of the formation and dissipation 
of eddies by viscous action. To take one example, they 
may be of importance in the theory of the aeroplane 
compass. The experiments of Part I. were undertaken 
to test the validity of these equations. 

§ 2. -Numerical Solution of Equation (7), and Discussion. 

The numerical solution of (7) is given in Table I. for the 
values of r/c and vt/c 2 which are there indicated. The values 
are probably accurate throughout to the fifth significant 
figure. For small values of the arguments, the value 
of <j>/Q sometimes differed from unity only after the sixth 
figure. The values of J 1 (« /i r/c) for arguments greater 

B2 



Mr. A. R. McLeod on Unsteady Motion produced 

Table I. 

Values o£ <j>/£l. 



l 


>t/c\ 


r/c = 0. 


•1. 


•2. 


•3. 


•4. 


•001 


1-00000 


1-00000 


1-00000 


1-00000 


1-00000 




002 


1-00000 


1-00000 


1-00000 


l-ooooo 


1-00000 




003 


1-00000 


1-00000 


1-00000 


1-00000 


100000 




004 


1-00000 


1-00000 


1-00000 


rooooo 


1-00000 




006 


1-00000 


1-00000 


1-00000 


1-00000 


1-00000 




010 


1-00000 


1-00000 


1-00000 


•99999 


•99992 




015 


rooouo 


1-00000 


•99996 


•99969 


•99795 




020 


•99991 


•99982 


•99935 


•99730 


•98970 




025 


•99907 


•99860 


•99650 


•99002 


•97242 




•030 


•99587 


•99451 


•98923 


•97591 


•94634 




•040 


•97490 


•97062 


•95616 


•92675 


•87504 




•050 


•92923 


•92196 


•89890 


•85665 


•79077 




•060 


•86299 


•85373 


•82526 


•77594 


•70409 




•070 


■78497 


•77482 


•74419 


•69279 


•62086 




080 


•70314 


•69294 


■66248 


•61234 


•54393 




•090 


•62300 


•61326 


•58440 


•53746 


•47444 




•100 


•54776 


•53877 


•51224 


•46943 


•41257 




■150 


•27221 


•26753 


•25321 


•23045 


•20079 




•200 


•13156 


•12916 


•12216 


•11104 


•09661 




•300 


•03034 


•02979 


•02817 


•02560 


•02226 




400 


*-0 3 6989 


•Q 2 6862 


•0 2 6489 


•0 2 5897 


•0 2 5128 




•500 


•0 2 1610 


•0 2 1581 


•0' 2 1495 


•0 2 1358 


•0 2 1181 




•600 


•0 3 3708 


■0*3641 


■0*3443 


•0*3128 


3 2721 




•700 


•0 4 8542 


•0 4 8386 


•0 4 7930 


•0 4 7206 


•0 4 6267 



vt/c 2 . 


r/c = -5. 


•6. 


•7. 


•8. 


•9. 


•001 


100000 


1-00000 


rooooo 


•99999 


•97034 


•002 


1-00000 


1-00000 


1-00000 


•99782 


86681 


•003 


1-00000 


1-00000 


•99982 


•98630 


•76998 


•004 


1-00000 


•99998 


•99865 


•96468 


69193 


. -006 


•99999 


•99944 


•98952 


•90552 


57793 


•010 


•99886 


•99004 


•94263 


•78057 


44051 


■015 


•98921 


•95570 


•85956 


•65615 


34305 


•020 


•96580 


•90409 


•77540 


•56140 


28162 


•025 


•93062 


•84548 


•69887 


•48810 


23855 


•030 


•88785 


•78592 


■63137 


•42973 


20630 


•040 


•79277 


•67418 


•52033 


•34229 


16189 


•050 


•69768 


•57707 


•43399 


27956 


12940 


•060 


•60940 


•49443 


•36541 


•23215 


10650 


•070 


•53008 


•42430 


•30983 


•19499 


08892 


•080 


•45994 


•36468 


•26406 


•16509 


07497 


•090 


•39845 


•31382 


•22589 


•14057 


06265 


•100 


•34483 


•27031 


•19375 


•12017 


0.H30 


•150 


•16624 


•12909 


•09175 


•05653 


02544 


•200 


•07985 


•08190 


•04393 


•02704 


012155 


•300 


•018395 


•014256 


•010114 


•0 2 6223 


022797 


•400 


•0 2 4237 


•0 2 3284 


•0 2 2330 


■0*1433 


3 6443 


•500 


•0 3 9760 


•0 3 7563 


•0 3 5366 


•0 3 3301 


3 1484 


•600 


•0 3 2248 


•0 3 1742 


•0 3 1236 


•0 4 7605 


4 3418 


•700 


•0 4 5178 


•0 4 4013 


•0 4 2847 


•0 4 1752 


5 7874 



* -0-0989 means -C06989. 



by Change in Angular Velocity. 5 

than 15 were obtained by the use of the first three terms 
of the asymptotic expansion. For smaller arguments the 
values were found by interpolation from the twelve-figure 
tables in Gray & Mathew's ' Treatise.' Curves representing 
Table I. are shown in fig. 1, from which figs. 2 and 3 
have been derived graphically. The calculation was made 



1-0 
■9 
■8 
7 


\m 












Fig.S. 




\\ 




















ft 




















6 


w 


v \ 






M) 


r^ — 










1 


\ ^ 


















\ 


\ 






\\ 










\ 


\ 




\ IK ^ 


K XvC 








V 






4 V X 


\\vV 


















L O^M 












8 \ 


X 




\\\^ 










%-^s 








p s ^^7 v ^^^r >, ^v^ 













"^h^*«-*L, "i" — f^^irr*'""- 





O .02. 04 .06 



.12 J 4- .Ut. .16 J8 .90 




with the aid of Chambers's seven-figure logarithms. It 
was checked throughout, once everywhere and twice or 
three times in parts. 

For large values of t, the angular velocities are pro- 
portional to Ji(ai?')/r {0 <r<_l). Most of the experimental 
curves are roughly of this form for a large value of t. 

The solution, being non-dimensional, applies to all sizes 



6 Mr. A. E. McLeod on Unsteady Motion produced 

of cylinder, all angular velocities X2, and all incompressible 
fluids. The significance of vt/c 2 is of interest. According 
to (6) and (7), the behaviour of a cylinder of liquid is 
exactly similar to that of a cylinder of double the radius 
in four times the time. Larger eddies should therefore 
be of relatively longer duration than smaller eddies. If 
in these same cylinders we have different liquids, the liquid 
in the larger cylinder must have a viscosity four iimes that 
of the liquid in the smaller cylinder, in order that (f> may 
have the same value for the same values of X2, r/c, and t. 
For air i> = 0*14, and for water ^ = 0'011 at ordinary tempe- 
rature, the ratio being about 12. Ignoring the compres- 
sibility of air, eddies of the assumed kind should die away 
much faster in air than in water, the values of c and 12 being 
the same. 

According to (6) and (7), the rates of growth and decay 
of (£/X2 in the cylinders are independent of 12. We shall 
see from the experiments that this is true only when 12 is 
very small, or when the radius c is a small fraction of the 
length of the cylinder. 

§ 3. Experiments. 

In the experiments which have been made, a brass 
cylinder, bored as accurately as possible and at least 
two diameters in length, was rotated about a vertical axis 
at constant speed. The cylinder was filled to within 1 cm. 
of: the top with ordinary tap-water ; and the observations 
consisted in timing through a measured angle by stop- 
watch, lycopodium particles floating on the surface, and 
so deducing the angular velocity <jf> at the radius r/c 
selected. The time of the stop-watch observation was also 
noted on a watch, in order to get the value of t from the 
instant of starting or stopping the cylinder. To time the 
lycopodium particles with more accuracy, a horizontal plane 
glass plate was mounted just over the cylinder. This was 
marked with ink in circles centring on the axis of rotation. 
The radii of the circles had the values r/c =0*3, 0*5, 0*7, and 
0*9, and a fifth circle was added at r/c= l'O to aid in centring 
the plate over the cylinder. Straight lines, 45° apart, through 
the centre of the circles served to indicate the angles through 
which the particles were timed. A large plane mirror, 
inclined at 45°, was placed on the glass plate to enable 
observations to be made from the side. 

To obtain freedom from vibration, the driving-motor and 



by Change in Angular Velocity. 7 

reduction gears were mounted on a separate table. The 
cylinders were clamped against a spinning-table which 
rotated very freely and easily in ball-bearings. Slight 
want of truth in the centring of the upper end of the 
cylinder was corrected by hanging a small weight on 
the rim, or by means of adjusting screws which worked 
in the rim of: the water-bath and pressed against the 
cylinder. 

Water-baths, providing a 2-inch layer of water around 
the sides and base, were fitted to each of the cylinders 
to make the temperature changes less. These rotated 
with the cylinder. In the case of: the larger cylinders 
especially, it was found that without a bath it was im- 
possible to secure uniform rotation of the water. When 
the room temperature was rising in the mornings or early 
afternoons, inward convection currents at the surface 
carried an excess of angular momentum towards the axis, 
and the angular velocity at the centre was sometimes for 
several hours 30 per cent, greater than that of the cylinder 
itself, the latter being very small. When cooling down, 
the effect is reversed and the "core" rotates slower than 
the cylinder. The water-baths made a great improvement, 
but with the largest cylinder, especially, the water in the 
bath and in the cylinder had to be mixed thoroughly before 
each experiment ; and the temperature of the water was 
regulated with an electric heater to be about o, l C. in 
advance of the room temperature (when rising). Some 
experiments lasted for nearly an hour, and a difference of 
o, l 0. between the temperatures of room and water was 
enough to start thermal currents having considerable effects 
(cumulative) on the observations. 

The cleaned surface of the water was lightly dusted with 
lycopo Hum, " rafts " not being allowed to form. When the 
temperature was being adjusted with the electric heater, 
the water was thoroughly mixed by stirring and bubbling air 
through it. The mean temperature was obtained for each 
experiment to 0°-01 C. 

At the end of each observation the stop-watch reading 
(given to 0*1 sec.) was recorded with the angle through 
which the lycopodium particle had rotated, and the time of the 
end of the observation as indicated by watch was also written 
down. A.11 observations in one experiment were made on a 
selected circle. The cylinder was timed frequently to verify 
the speed, which was constant to within a per cent, or two, 
and irregularities were corrected by an adjustable resistance. 



8 Mr. A. R. McLeod on Unsteady Motion produced 

The motor was driven from a 200-volt accumulator circuit. 
Three sizes of cylinders were used : 

Small cylinder c = c x = 2*40 cm. = radius. 

Middle „ c = e 2 = 7'48 „ 

Large „ c = c 3 = 12*69 „ 

Three angular velocities were selected for each cylinder : 

Low speed £2 = 12! = '1396= 1^ r.p.m. 

Middle „ n=n 2 =ro47 =10 „ 

High „ O = H 3 = 3-770 =36 „ 

The large and middle cylinders were painted inside. 
They were bored with an error of a small fraction of a 
millimetre at the upper rim. The small cylinder was true 
to 0'05 mm., and its length was four diameters. The others 
were slightly more than two diameters long. The bases of 
all were plane and smooth on the inside. 

Experiments were made when the water began to rotate 
from a condition of rest, and also when it was coming to 
rest from a condition of uniform rotation, the uniformity 
being ascertained before the cylinder was stopped. 

In reducing the observations, the watch- reading was 
corrected by subtracting half the stop-watch reading to 
give the mean time t, and the values of vtjc 2 and 0/X2 were 
calculated for each observation. The viscosity of the water, 
at the mean temperature during the experiment, was taken 
from Kaye & Laby's Tables. The values of 0/12 were 
plotted on squared paper against the values of vt/c 2 , each 
observation being represented by a dot. Hence for a 
selected circle corresponding to the selected value of r/c, 
for each value of 12 there corresponds a series of dots 
which lie in a narrow band. This band defines a curve 
by its median line. Observations were made on the four 
circles corresponding to r/c = 03, 0*5, 0*7, and 9 for each 
of the three cylinders. Thus for each cylinder there were 
finally eight sheets of curves — four for " starting " and 
four for " stopping- " experiments. Each of these sheets 
contained three curves corresponding to the three selected 
velocities, and corresponded to one of the values of r/c. 
The theoretical curve from Table I. was also added for 
comparison. Several experiments were made for each set 
of conditions, to secure a sufficiently dense band of points 
for each curve. Apart from a number of preliminary 



by Change in Angular Velocity. 9 

experiments, the results given here are based on 215 distinct 
experiments. 

In taking the stop-watch readings, disturbances of the 
circular motion of the lycopodium owing to turbulence 
or eddying were avoided as much as possible, and selections 
were made of those motions which lay most nearly along 
the circle considered. This means that when turbulence 
is increased, the angles through which the particles are 
timed become less. The effect of the turbulence is merely 
to broaden the band of points, but the mean motion, repre- 
sented by the median line, is always well-defined. 

§ 4. Discussion of Results. 

The results obtained are embodied in the accompanying 
curves. These are derived from the observational curves 
mentioned in § 3. The following reference table explains 
the figures : — 

Stopping. 

Starting. 

Stopping. 

Starting. 

Stopping. 

Starting. 

Stopping. 

Starting. 

Stopping. 

Starting. 

Stopping. 

Starting. 

Stopping. 

Starting. 

Stopping. 

In these figures each curve represents the distribution of 
angular velocities </}/£! over a radius of the cylinder at 
the time vtjc 2 corresponding to the number given alongside 
the curve. The times in seconds corresponding to these 
numbers may be obtained by use of the following table, 
the water being at 17° C. : — 

c l = 2'40 cm. vtjc 2 = '01 corresponds to /= 5*30 sees. 
c 2 = 7-48 „ vt/c 2 = -01 „ „ t= 51-8 „ 

c 3 =12-69 „ vt/c 2 = '()l „ „ t= 148-4 „ 



.4. 


Cl =2-40 


cm. 


n 2 =10 r.p.m. 


5. 


•>•> 




I2 3 = 36 


6. 


?5 




£2 3 = 36 „ 


7. 


c 2 = 7*48 


cm. 


n t = n „ 


8. 


>y 




n,= ij „ 


9. 


55 




G 2 =10 „ 


10. 


;? 




n 2 =io „ 


11. 


5> 




n 3 =36 


12. 


?5 




X2 8 =36 


13. 


c 3 = 12-69 


cm. 


Gi= H » 


14. 


?i 




n,= ii „ 


15. 


?5 




n 2 =io 


16. 


•>y 




n 2 =io 


17. 


5? 




n 3 =36 


18. 


J? 




123 = 36" 



10 Mr. A. R. McLeod on Unsteady Motion produced 

Comparison with the theoretical curves (figs. 2 & 3) 
shows a marked departure in all cases but that o£ the 
slow speed and small cylinder. The agreement with 
theory improves as r—s~c, but there is still a large departure 
at r/c = '9 for the large and middle cylinders, except at the 
low speed. 




Vc w 



Again, it is noticeable that as the radius of the cylinder 
increases, the departure from theory becomes more marked. 
This might be expected as there is, near the axis, relatively 
less constraint from the boundaries with the larger cylinders. 
Accordingly, if we suppose an eddy of this kind rotating in 
a lake of stationary water, and if instead of stopping the 
cylinder wall we annihilate it, we expect the eddy to dis- 
appear more quickly than if the stationary solid wall had 



by Change in Angular Velocity. 11 

been retained, for the constraint will be still further reduced 
and greater irregularity is possible. 

It is particularly noticeable that as tlie constant rotation D. 
of the cylinder is increased, the departure from theory becomes 
more marked. 




Only three figures were obtained for the small cylinder, 
because the agreement with the theoretical curves in the 
three cases omitted will be practically exact. 

The curves for the " stopping " experiments show a much 
greater departure from theory than the curves for the 
"starting" experiments. This is due to the break-up of 
the regular motion, owing to instability at the fixed outer 
wall. Except with the high speed and the large cylinder, 



12 Mr. A. R. McLeod on Unsteady Motion produced 

the motion on starting, on the other hand, appears to the eye 
to be without appreciable irregularity, and it is very striking 
to see the sharp dividing-line between quiescent liquid in the 
centre and rotating liquid on the outside. This dividing-line, 
represented by the steep part o£ the curve in the figures, 
slowly moves towards the centre but becomes indistinct some 
distance from it. It is best seen at the higher speeds when 




the velocity-gradient is greater. Its rate of travel depends 
on the value of O. 

In the case of the large cylinder starting at the high speed, 
large secondary eddies 3*5 cm. across were often observed 
just inside the cylinder wall a few seconds after starting the 
motor. These soon died out when the velocity-gradient 
became less, and thereafter the motion travelled in towards 



by Change in Angular Velocity. 13 

the centre regularly. The effect of these eddies was observed 
to cause a shifting of the curve for <j>/£l in the direction of 
a greater <£ for the same value of vtjc 2 . In the same case, 
secondary eddies were observed about the "4 circle when the 
motion had reached the centre. Although mean velocities 
were recorded (median Hue of the band of points), the effect 
is shown in fig. 17 by the wavy appearance of tie two upper 
curves. 

When the cylinder is stopped, the water continues to 
rotate until the irregular motions, generated near the cylinder 
wall, have had time to extend inwards. Small eddies then 
travel about, and the central axis of rotation wanders con- 
siderably and often seems to disappear temporarily amid 
cross-currents. The motion is very irregular except at the 
low speed, and even in this case some irregularity al\\ ays 
remains. The lycopodium particles do not follow the circles 
for very long, and are usually moving at an angle to them. 

With the large cylinder at the high speed, the velocity 
immediately after stopping the cylinder seemed to give 
stability and to aid in preserving the circular character 
of the motion ; but when the kinetic energy had somewhat 
diminished, eddying became more noticeable. 

On the curves mentioned in § 3 in which <£/H is plotted 
against vtjc 2 , the bands of points are much narrower in the 
" starting " experiments than in the others, and determine 
the position of the median line easily to *001 in the value of 
vtjc 2 in most cases. For the " stopping" curves, the limit 
of error may be two or three times this occasionally. One 
noticeable effect is that the band is narrow when the velocity- 
gradient has a considerable value, i. e. when the curves in the 
figures slope steeply. In these cases considerable momentum 
is being transferred through the water, and there will be 
considerable shearing stress and vorticity, and the stability 
might therefore also be considerable. As soon as the 
velocity-gradient becomes small, the band of points broadens. 
For example, in the " starting " curves the bands are some- 
times very narrow until the value of <j>/fl has risen to 0*9, 
when they broaden out. Conditions seem to favour irre- 
gularity at the centre (axis) of the cylinder where the 
velocity-gradient vanishes. On the axis the stability is a 
minimum. In the "stopping" curves the bands are nar- 
rower the greater the angular velocity O, i. e. the greater 
the vorticity of the water, especially near the cylinder wall 
where the instability originates. 

The observational curves show that viscosity alone is not 



14 Unsteady Motion in a Rotating Cylinder of Water. 

sufficient to account for the effects, except for small values 
of c and Q, i. e. for long, narrow cylinders and slow speeds. 
A little dye introduced into the rotating water shows no 
signs of any minute eddying or micro-turbulence ; and so 
we must look for currents in the water as the cause, of the 
discrepancy, which is obviously the case in the " stopping " 
experiments. The formation of large eddies in large bodies 
of fluid seems to be due chiefly to the interaction of two 
local currents, or to low pressure caused by an obstacle or a 
sink, and not to the slower processes of viscosity. 

If we attribute the deviation from theory to an ignored 
increase in the kinetic viscosity v, we find that when the 
large cylinder is stopped at the high speed, the increase 
would have to be represented by a factor exceeding 10 in 
value nearly everywhere, while the value would lie between 
50 and 100 on r/c = 0'3 shortly after stopping. With the 
middle cylinder, stopped at the high speed, the factor has 
about half these values ; and with the small cylinder, stopped 
at the same speed, the factor ranges from 1*3 to 3*0. In 
the starting experiments the factors are nearly unity, but 
they are meaningless here as the motion is not turbulent. 

Some earlier experiments illustrate the instability of the 
stopping experiments. In these an inner cylinder was 
rotated coaxially with a fixed outer one. As is well known, 
it was found that at no speed of rotation of the inner cylinder 
was it possible to set the water moving in circular paths, o wing- 
to the eddies which were continually thrown off. The slower 
the speed of rotation the more conspicuous were the eddies, 
especially on the borders of the outer, more slowly-moving 
water. Measurements of the angular velocity showed a large 
departure from theory, the inner parts rotating more slowly 
and the outer parts more rapidly than the theory indicates. 
The effect o£ the travelling eddies is thus to make the angular 
velocity more like that of a rigid body. When the speed was 
very great (2500 r.p.m.) the kinetic energy seemed to give 
stability to the water. A whirlpool formed next the inner 
cylinder, and a large oscillation was presently set up in the 
form of a wave with its crest along a radius of the outer 
cylinder and its trough on the other half of the same diameter. 

Some thick, very viscous oil residues, when rotated in a 
cylindrical tin about 15 cm. in diameter, acquired the full 
velocity on starting (36 r.p.m.) in something less than 
4 seconds, and came to rest in the same time when the 
cylinder was stopped. Only a slight displacement of the oil 
occurred, the surface being momentarily roughened with fine 
lines like cracks. 



[ 15 ] 

II. The Elements of Geometry. 
By Norman Campbell, Se.D* 

Summary. 

IT is maintained that the geometry of Euclid is best 
interpreted as an attempt to deduce as many important 
propositions as possible from the assumption that length, 
angle, area (and perhaps volume) are magnitudes uni- 
versally measurable by the methods that are actually 
employed in experimental physics. All his chief pro- 
positions (in so far as they are true) can be deduced from 
that assumption without any other. 

This view is supported, not by a detailed analysis of 
the Elements, but by a very summary sketch of the laws 
that must be true if the assumption is to be acceptable. 
In a sequel it is hoped to discuss similarly the foundations 
of another branch of experimental geometry with which 
Euclid is not directly concerned — namely the geometry of 
position, which involves the concept of " space." 



1. There was formerly much discussion whether geometry 
was an experimental or a mathematical science. It is now 
generally agreed that there are two closely connected 
sciences, one mathematical and one experimental. The 
former, which has been defined as the study of multi- 
dimensional series, consists of a logical development of 
ideas which have no necessary dependence on the experience 
of the senses. It does not consist of laws and cannot be 
proved or disproved by experiment ; it can enter into 
relation with experimental science only through theories 
and by suggesting hypotheses which, interpreted suitably, 
predict laws. The formulation of such theories, in which 
Minkowski was the pioneer, is one of the most striking 
features of modern mathematical physics. The experi- 
mental science, on the other hand, is meaningless apart 
from experience, and its propositions are true or false 
according as they agree or disagree with experiment. 
They are the very fundamental laws which involve only 
the geometrical magnitudes such as length, angle, or area. 
It may be noted in passing that the laws predicted by 
geometrical theories are not in general geometrical laws, 
but involve electrical, optical, or dynamical concepts. 

* Communicated by the Author. 



16 Dr. Norman Campbell on the 

The mathematicians who have recently taken over from 
the philosophers the task of teaching experimenters their 
business have decided that only the mathematical science 
is properly termed geometry. In support of their claim 
they appeal to the authority o£ the Greeks, and thereby 
imply that Greek geometry is mathematical and not experi- 
mental. This implication raises questions of scientific 
interpretation and not of mere convenience in nomenclature. 
For the matter cannot be decided by inquiring what Euclid 
(for example) thought he was writing about : it is admitted 
that, as an exponent of mathematical geometry, he was 
guilty of errors ; and, if he was capable of error, he may 
have been wrong as to the nature of his assumptions and of 
his arguments. If we are justified today in confining the 
term to one study rather than another, because that term 
was used by Euclid, it can only be on the ground that 
Euclid's propositions and his methods of proving them are 
closely similar to those employed today in that study. 

If this test is applied, geometry is an experimental 
science. For whereas the Elements is utterly different 
from anything modern mathematical geometers produce, 
it is, judged by modern standards, quite a creditable 
attempt at an exposition of experimental geometry. It 
can be regarded broadly as an attempt to deduce as many 
important laws as possible from the single assumption 
that length, area, angle, and (less definitely) volume are 
magnitudes, universally measurable by the methods which 
are actually employed in experimental physics, or to which 
the methods that are actually employed would be referred 
if doubt arose concerning their validity. Nothing is assumed 
but that every straight line has a length, every pair of straight 
lines an angle, and every plane surface an area. The 
definitions, axioms, and postulates should then be state- 
ments of the laws by virtue of which measurement is 
possible. It is admitted that the attempt is not wholly 
successful ; but its faults, or many of them, are readily 
explicable : the author has not to be represented (as he 
must be if he is an exponent of the mathematical science) 
as constantly straining at gnats and swallowing camels. 

Such a view can be established only by a detailed and 
tedious criticism which, in so far as it concerns Euclid's 
intelligence, is not of scientific interest. In place of it 
will be offered a very summary sketch of the fundamental 
notions and laws of experimental geometry and sufficient 
comparison of them with Euclid's assumptions to suggest 
that on them might be founded a deduction, by methods 



Elements of Geometry. 17 

very similar to those that he employs, of the propositions 
which he actually states. References are throughout to 
Todhunter's edition. 

(2) But two preliminary questions must be asked. F-irst, 
can an experimental science be deductive at all? Certainly 
it can. A deduction from a law is an application of that 
law in particular circumstances which were not examined 
when it was formulated. If, after examining the sides of 
squares and of triangles, I assert the general law that 
all straight lines have measurable lengths, and then, without 
further experiment, assert that the diagonals of squares, 
which are also straight lines, are also measurable, I am 
miking a deduction. It may be true that there is some- 
thing precarious about the results of such deduction — 
that question is not raised here, — but the deduction itself 
is quite unexceptionable ; the falsity of the conclusion is 
definitely inconsistent with the truth of the premises. 
If deubt is raised concerning the conclusion, the ultimate 
means of resolving it is by experiment ; but experimental 
science, in the hands of its greatest exponents, consists in 
asserting- such general laws that doubt does not arise 
concerning the results of deduction based on them. 

The second question is whether there are truly laws 
which make measurement possible. The question is dis- 
cussed at length in my ' Physics/ Part II., the results and 
nomenclature of which will be used freely in what follows. 
But there is one matter which may receive special mention 
here, because it is concerned with " incommensurables," 
which are often (but falsely) believed to be of especial 
importance in geometry. Measurement is possible when, 
by means of definitions of equality and addition, a standard 
series of the property in question can be established, starting 
from some arbitrary unit, such that any system having the 
property is equal in respect of it to some one member of 
the standard series. Now (it might be argued) such mea- 
surement is not possible for length, because the diagonal of 
a square cannot be equal to any member of a standard series 
based on the side as unit ; indeed that result is actually 
proved by Euclid. Consequently it is patently absurd to 
pretend that Euclid's propositions can be derived from an 
assumption, namely that measurement is possible, which is 
inconsistent with its conclusions. 

One method of escape from this difficulty may be 
mentioned, although it will not be adopted. A slight 

Phil. Mag. S. 6. Vol. U. No. 259. July 1922. 



J 8 Dr. Norman Campbell on the 

amendment in the thesis might be made, and it might 
be said that Euclid's assumption is that the laws are true 
which would make measurement possible if there were no 
incommensurable lengths — for these laws, though necessary 
to measurement, may not be sufficient. But the difficulty 
vanishes entirely, if it is remembered what is meant by 
" equality " in experimental measurement. When it is 
said that A is equal to B, it is meant that there is no 
possible means of deciding which of the two is the greater. 
If then I say that the diagonal of a square is \/2 times 
the side, I mean that, if I measure the diagonal in terms 
of the side as unit, there is no means of deciding whether 
the value obtained, when multiplied by itself according to 
the multiplication table, will be greater or less than 2. 
That statement is not in the least inconsistent with my 
assigning to particular diagonals values of which the square 
is not 2 ; it is only inconsistent if a law can be found 
by which I can tell in particular cases whether the square 
will be greater or less than 2. My assertion is that* there 
is no such law ; and that assertion is true. In its appli- 
cation to all magnitudes except number, equality must be 
interpreted in this, slightly statistical, sense. 

3. There is then no preliminary objection to the view 
that Euclid's propositions are deductions from the laws in 
virtue of which the geometrical magnitudes are measurable. 
We now proceed to ask what those laws are. 

Geometrical conceptions are derived ultimately from 
our immediate sensations of muscular movement, just as 
dynamical conceptions are derived from our sensations of 
muscular exertion and thermal conception from our sense 
of hot and cold. We have an instinctive and indescribable 
appreciation of differences in direction of various movements ; 
we appreciate that one direction may be between two others ; 
and if other sensations (e. g. those of hot and cold or rough 
and smooth) vary with movement along a certain direction, 
we appreciate that of the varying sensations some are between 
others. The notions of direction and of the two kinds of 
betweenness are the foundations of geometry. It is a vitally 
important fact that there is an intimate relation connecting 
betweenness determined by one kind of muscular motion (e. g. 
that of the hand) and that determined by another {e.g. that 
of the eye). The relation is much too complex for any 
account of it to be attempted here ; but it is only because it 
exists that "space" explored visually or by our different limbs 
is always the same. 



Elements of Geometry. 19 

The fundamental notions give rise to those o£ surfaces and 
lines. Surfaces are connected with the fact that a sensation 
may be unaltered by movement in any of a certain group 
of directions (which are said to be in a surface and to cha- 
racterize it), while it may be altered by any movement in 
any direction not in this group (directions away from the 
surface) . Of lines there are two kinds, which will be termed 
respectively " edges" and " scratches."" Edges arise from 
the fact that the group of directions characteristic of a 
surface may change suddenly at some part of it. It is 
a matter of convenience whether the parts characterized by 
different directions are spoken of as different surfaces or 
as parts of the same surface : we shall adopt the second 
alternative. Scratches arise from the fact that, while the 
directions characterizing a surface are unaltered, the sen- 
sation the occurrence of which distinguishes "in the 
surface" from "out of the surface" may change suddenly. 
Some, but not all lines, are such that the whole of them 
lies along a single direction. Points are of little importance 
in the earlier stages of geometry ; they arise from the fact 
that two lines may have a part in common. Two points, 
both on the same line, are termed the ends of the part of 
the line between those points. 

The recognition of surfaces and lines is the first step 
towards geometry. Euclid attempts to give an account 
of them in Defs. 1, 2, 3, 5 of Book I., which are the least 
successful part of his treatise. The account given of them 
here is no better than Euclid's for the purpose of conveying 
a notion of them to one who does not possess it already ; 
but since there are no such persons, the objection is not 
serious. But our account is better in drawing attention to the 
notions that are fundamental in geometry and in not assuming 
familiarity with conceptions, such as length, which are 
necessarily subsequent. 

4. Some surfaces, but not all, when subjected to muscular 
force undergo only such changes as can be compensated by 
a suitable movement of the whole body ; if such a movement 
is made, the group of directions characterizing the surfaces 
is restored. In other words, such surfaces can move without 
alteration of form ; they provide the original and crude con- 
ception of a rigid body. By means of the motions of rigid 
bodies, it is sometimes possible to bring parts of two pre- 
viously distinct surfaces info contiguity, so that (here is 
nothing between those parts. In particular, edges, or parts of 
edges, can often be brought into such contiguity. Scratches 

02 



20 Dr. Norman Campbell on the 

can be brought into contiguity with edges, and, in a sense, 
into contiguity with other scratches ; but the criterion of 
contiguity in the last case is much less direct and requires 
methods involving something other than the simple per- 
ception of nothing between. 

The recognition of the possibility of contiguity is the 
second step towards geometry and leads immediately to 
the third, which consists in the establishment of a definite 
criterion for a straight line. A crude criterion is provided 
by direct perception : a young child knows the difference 
between a straight and a bent line by simply looking at 
them; the recognition seems to depend on the fact that 
a straight line is all in one direction and is symmetrical 
with regard to the unsymmetrical directions of left and 
right or back and front. The crude criterion is stated as 
well as it can be in Euclid's Def. 4. But contiguity 
provides a much more stringent criterion, which in the 
first instance is applicable only to edges and not to 
scratches. Two edges are straight if, when two portions 
of one are brought into contiguity with two portions of 
the other, all the portions between these two portions are 
also in contiguity, however the contiguity of the first pairs 
of portions is effected. It appears as an experimental 
fact, that if A, B and 0, D are two pairs of straight edges 
according to this criterion, C is also straight if tested 
against A ; accordingly an edge can be called straight 
independently of the other member of the pair on which 
the test is carried out. A scratch is straight if it can be 
brought into complete contiguity with a straight edge, 
Those facts are stated in Axiom 10. 

Other definitions of a straight line are sometimes offered : 
e. g., (1) an axis of rotation, (2) the shortest distance 
between two points, (3) the path of a ray of light. (1) is 
almost equivalent to that stated here; (2) will be noticed 
presently ; (3) is not accurately true (i. e., if it is adopted, 
the familiar propositions about straight lines are not true), 
but it is important as an approximation for comparatively 
rough measurements. 

A plane surface (or, according to our usage, part of 
a surface) is then defined as in Def. 7. It can also be 
defined by the complete contiguity of three pairs of sur- 
faces ; but the contiguity of surfaces is not easy to describe 
accurately. Such a definition is, however, actually used in 
making optical flats and surface plates ; if it were adopted, 
it would still be necessary to introduce the fact that it 
agrees with our definition, in order to measure angle. The 



7<?r 



Elements of Geometry. 21 

conception of the contiguity of surfaces is not actually 
required, except perhaps for the measurement of volume. 
{Cf. § 11.) 

5. The third step places us in a position to introduce 
measurement and the three fundamental magnitudes, 
length, angle, and area. For fundamental measurement 
we need definitions of equality and addition, such that 
the law of equality and the two laws of addition are 
true. The choice of unit may be left out of account ; for, 
with geometric magnitudes, the laws are true whatever 
unit is selected. The law of equality is Axiom 1; the first 
law of addition is Axiom 9. Axioms 2-7 are together very 
nearly equivalent to the second law of addition (which may 
be stated roughly in the form that the magnitude of a sum 
depends only on the magnitudes of the parts). Axiom 8 is 
an attempt to compress the definitions of equality for all 
three magnitudes into a sinole sentence ; it is better to 
separate them. Euclid fails to give any definition of 
addition ; he does not tell us how the " w 7 hole " is to be 
related to the "parts" in order that it should be greater. 

6. We will now take the magnitudes in turn. For the 
length of a straight line the necessary definitions are : — 

(1) Two straight lines are equal in length if they can be 
placed so that when one end of the first is contiguous with 
one end of the second, the other ends are also contiguous. 

(2) The length of the straight line AB is equal to the sum 
of the lengths of the straight lines CD, EF, if they can be 
placed so that C is contiguous with A, F with B, I) with E 
and with some part of AB between A and B. 

These definitions, like all similar definitions of mag- 
nitudes, are satisfactory and are subject to the necessary 
laws of equality and addition only if certain conditions 
are ful rilled. The conditions are described by saying that 
the surfaces in w T hich the straight lines lie must be those 
of rigid bodies. This is a definition of a rigid body: a 
rigid body is something which (like a perfect balance) is 
determined by the satisfaction of the conditions for mea- 
surement *. Rigid bodies according to this test include 
many of those which satisfy the crude test of § 4, though 
they include others (e.g., surveyors' tapes used as surveyors 
use them) which do not satisfy that test. In virtue of the 
fact that rigid bodies are necessary to measurement, the 

* Cf. H. Dingier, Phys. Zeit. xxi. p. 487 (1920). 



22 Dr. Norman Campbell on tlie 

branch of geometry with which we (and, according to our 
view, Euclid) are concerned may be fitly described as the 
study of the surfaces of rigid bodies. It is thus dis- 
tinguished from a wholly different branch of geometry, 
with which we are not here concerned, ihat is not confined 
to rigid bodies ; this is the geometry of position. 

It is important to notice that not all pairs of straight 
lines can be brought into contiguity, and that the law of 
equality cannot therefore be tested universally. It might 
have turned out that there was some material difference 
between those which can and those which cannot be brought 
into contiguity with a given line ; and that if we assumed 
that the law of equality is universally true, we should be led 
to inconsistencies. It is an experimental fact that no such 
inconsistencies do arise when we extend our definition of 
equality so that lengths are equal when they are equal to 
to the same length, although they cannot be brought into 
contiguity with each other. This is, of course, one of the 
most important laws that make measurement possible. A 
similar remark applies to all the geometric magnitudes and 
need not be repeated. 

7. The length of lines that are not straight can be 
measured approximately as fundamental magnitudes by 
means of flexible but inextensible strings. But the laws 
of such measurement are not strictly true, because (as we 
say now) no string is infinitely thin and the surface never 
coincides with the neutral axis. Another possible way, 
perhaps more accurate but of limited application, would 
be to roll curved edges on some standard edge, which 
need not be straight. But in truth there is no perfectly 
satisfactory way of measuring fundamentally the length of 
curved lines. All the measurements which we make on them 
are derived from measurement of straight lines ; they involve 
numerical laws between fundamentally measured magnitudes. 
One of these laws is that the perimeters of the circumscribed 
and inscribed regular polygons tend to a common limit as 
the number of sides is increased. That law is therefore a 
law of measurement if curved lines are to be measured. 

The question whether curved lines can be measured 
fundamentally is important, because, if they could be, it 
would be possible to define a straight line as the shortest 
distance between two points. (The definition would have 
to be put in some other form, since distance, a conception 
belonging to the geometry of position, implies the mea- 
surement of length.) But since they cannot be, that 



Elements of Geometry. 23 

definition must be rejected ; it must be regarded merely as 
a generalized form of Prop. I. 20. 

8. Angle is the measure of the crude conception of 
direction. The following are the definitions of equality and 
addition for the angle between two intersecting straight 
lines : — The angle between two straight lines A, B is equal 
to that between C, D if it is possible to bring A into con- 
tinguity with C and B with D. The angle between A, B is 
the sum of the angles between C, D and E, F, if, when 
A is brought into contiguity with C and D with E, I) lying 
between C and F and in the same plane with them, F can 
be brought into contiguity with B. These definitions are 
satisfactory only if the straight lines are in rigid bodies; 
or, in other words, there are surfaces which satisfy the 
conditions for the measurement of length and also those 
for the measurement of angle. 

But even if the surfaces are those of rigid bodies, the 
definitions are not wholly satisfactory and the laws of 
measurement not entirely true. We must distinguish 
angles according as the two straight lines which they 
relate are or are not prolonged on both sides of the 
common point : the latter class may be termed " corners/' 
the former " crossings." Angles between edges are always 
corners ; those between scratches may be either corners or 
crossings. If we try to include both corners and crossings 
in the same class as a single magnitude, the law of equality 
is not true ; for two corners which are both, according 
to the definition, equal to a crossing may not be equal 
to each other ; as we say now, one angle may be the 
supplement of the other. But if we treat corners and 
crossings as separate magnitudes this difficulty disappears ; 
the law of equality is true for either taken apart from the 
other. Actually we take corners only as magnitudes ; 
crossings we measure by the corners with which they can 
be made contiguous. Each crossing then has four angles 
(i. e. corners) associated with it. It is an important experi- 
mental fact that the " opposite" angles are equal; it is 
best taken as a primary law, instead of being proved from 
other axioms as in Prop. I. 15. It is a law of measurement, 
because if it were not known, we should need four and not 
two angles to measure a crossing ; it is thus inherent in our 
system of measurement. 

But though the law of equality is now true, the first law of 
addition is false ; it is false for both corners and crossings. 
The whole which is the sum of the parts may be equal to 



24 Dr. Norman Campbell on the 

one of the parts : e. </., if both of two parts, being corners, 
are what we now call 120°. Some kind of spiral space can 
be imagined in which the law would be true ; but actually 
it is very important that it is false. For, apparently in- 
separable from its falsity is the fact that the angle between 
two portions of the same straight line can be measured and 
given a finite value in terms of a unit which is the angle 
between two intersecting lines. This fact is described by 
the assertion that there are right angles and that a per- 
pendicular can be drawn to any straight line from any 
point in it, a right angle being defined as in Def. 1. 10. 
(Axiom 1. 11 follows from this definition, regarded as an 
existence theorem, and our axiom Prop. I. 15.) Since the 
existence of right angles is vital to geometry, we cannot 
avoid the falsity of the first law of equality by some 
alteration of the definition. We can only recognize that 
the law is true in some conditions, and be careful to apply 
it in deduction only when it is true. It is true when all the 
lines making the added angles lie on the same side of (or 
contiguous with) a single straight line passing through 
their common point ; this condition can be expressed, 
though with some complexity, in terms of the fundamental 
notion of between. Thus, in proving Prop. I. 16 we need 
to know that OF and CD both lie on the same side of AC 
This law, and perhaps others of the same nature, are laws of 
measurement, defining the conditions in which angle can be 
measured uniquely. They require explicit mention. 

The ambiguity which the falsity of the first law of 
addition introduces into numerical measurement is removed 
by certain conventions. These need not be considered here 
for we are not assigning numerical values. 

If the length of curved lines were measurable funda- 
mentally, angle might be measured as a pure derived 
magnitude, e. g. by the ratio of the arc to the radius of 
a circle in virtue of the numerical law, established experi- 
mentally, that the arc is proportional to the radius. But 
since curved lines cannot be so measured, we must take 
angle to be fundamental. We cannot use right-angled 
triangles with straight sides to measure angle as derived, 
because we need fundamental measurement to determine 
what angles are right. Of course we might define for 
this purpose a right angle as an angle between some two 
lines arbitrarily chosen as standard ; but such measurement 
would be intolerably artificial and nothing whatever could 
be deduced from such a definition. 



Elements of Geometry. 25 

9. Euclid's definition of parallel lines must be rejected 
entirely, for, since all plane surfaces are limited, the 
criterion suggested is inapplicable. Since the crude de- 
finition of parallelism is similarity of direction, we may 
try to define parallel lines as those which being in the 
same plane make the same angle with any third line. 
We thereby imply the axiom of parallels in the form 
(Prop. I. 29) that such lines which make the same angle 
with one straight line make the same angle with any other ; 
we imply also that the angles which are to be equal are the 
"exterior" and "interior" opposites or the "alternate" 
angles, since if the interior angles are compared the 
proposition is not true. But the definition is not very 
satisfactory; for, when the lines are edges, there is not 
always an exterior or an alternate angle. It is better 
to adopt the substance of Axiom I. 12 as a definition, 
and to say that lines in one plane are parallel when the 
sum of the interior angles is equal to two right angles. 
This much abused axiom seems to me a very ingenious 
way out of a real difficulty. We then assert the axiom of 
parallels in the form (implied by I. 32) that if any two 
straight lines in a plane are cut by any third line, the 
sum of the interior angles is the same for all third lines. 
The merit of this axiom is that it indicates clearly that the 
" axiom of parallels " is really something concerning all 
straight lines in a plane and not only parallel lines, and 
that parallel lines are merely a particular case of other 
pairs of lines. The propositions that parallel lines never 
do intersect and that the angle between them is zero follow 
immediately. 

The axiom of parallels is a law of measurement because 
it is involved in the measurement of the angle between lines 
w r hich do not intersect. Its use for this purpose requires 
that at some point of a straight line it should always be 
possible to place a straight line parallel to a given straight 
line. This proposition is not true for concave surfaces, but 
the complexities arising from this failure and the means of 
avoiding them may be left for the present ; they are dealt 
with more naturally in connexion with "space." If the 
axiom were not used, we could not by our present methods 
measure the angle between non-intersecting straight lines : 
first, because the definition of equality given above, though 
sufficient for such lines, is not necessary : second, because 
the definition of addition is wholly unsatisfactory. 

There has been so much discussion of the necessity of the 



26 Dr. Norman Campbell on the 

axiom of parallels that the matter requires rather more con- 
sideration. Two questions are involved. First, would it 
be possible to measure the angle between non-intersecting 
lines without assuming some proposition logically deducible 
from the axiom? It would be if, and. only if, some 
property, common to all lines between which the angle 
is the same, can be found which is determinable by direct 
experiment not involving parallel lines. There may be 
such a property, but I have not been able to think of it. 
Second, if the axiom were not actually true — but we may 
stop there. In a pure experimental science, there is no 
sense in asking what would happen if the world were other 
than it actually is. Theory is necessary to give such a 
question a meaning, by suggesting what might remain 
unaltered during the change. For our present purpose 
the axiom is as necessary as any other of those we are 
considering. 

10. Area is distinguished from all other fundamental 
magnitudes because the definitions of equality and addition 
are inseparable. They may be expressed thus. The areas 
of two bounded plane surfaces are equal if (but not only if) 
their boundaries can be brought into complete contiguity 
with each other or with the same third boundary. (A 
bounded surfaca is a part of a surface which includes all 
portions which can be traversed without crossing the 
boundary line.) The area of A is the sum of the areas 
of B and C, if when parts of the boundaries of B and C 
are brought into contiguity with each other, the remaining 
parts of the boundary can be brought into contiguity with 
the boundary of A. In virtue of the fact that parts of the 
boundaries of two surfaces can be brought into contiguity 
in many different ways, there may be many different 
bounded surfaces, of which the boundaries cannot be made 
contiguous, which are the sum of the same bounded sur- 
faces. If the measurement of area is to be satisfactory, 
these surfaces must also be deemed to have equal area, and 
the definition of equality must be extended correspondingly. 
With this extension the laws of equality and addition are 
true, and the measurement is satisfactory. 

In order that all bounded plane surfaces should have 
areas, some rule must be found for choosing the shape of 
the members of the standard series and for grouping them 
in such a way that some sum of them is equal to any area. 
We use for this purpose rules based on the axiom of 
parallels, and that axiom is therefore again a law of the 



Elements of Geometry. 27 

measurement of area. The rule might possibly be dis- 
pensed with, it we were prepared to spend unlimited time 
in selecting by trial and error shapes for the members of 
the standard series which fulfil the necessary conditions ; 
but actually we could never measure area except by making 
use of similar figures, the production and properties of 
which depend wholly on the axiom of parallels. Further, 
it is the use of that axiom which enables us nowadays to 
calculate area from the linear dimensions of a surface 
without resorting at all to fundamental measurement. 
But of course all the numerical laws on which that cal- 
culation depends have to be established by means of 
fundamental measurement. It is only by defining area 
as we have done, and assuming the axiom of parallels, 
that we can prove by deduction that the area of a rect- 
angle is proportional to the product of its sides, or equal if 
the units are suitably chosen. 

The areas of surfaces that are not plane cannot be mea- 
sured fundamentally, even to the extent that the length of 
curved lines can be. For there are no inextensible surfaces 
which can be brought into contiguity with surfaces of any 
curvature. Measurement of curved area is always derived 
and estimated by the limit of the circumscribed polyhedra 
as the number of their sides is increased. But the whole 
matter is obscure, because it is much more difficult to 
establish experimentally that there is a limit or to say what 
the limit is ; for there is here no inscribed polyhedron 
tending to the same limit. There is singularly little experi- 
mental evidence for the assertion that the area of a sphere 
is 47rr 2 , and there is great difficulty in saying exactly what 
we mean by such an assertion ; curved area is almost always 
a hypothetical idea and not an experimental magnitude 
at all. 

11. Volume is a property of complete surfaces. Since 
complete surfaces can never be brought into complete con- 
tiguity, volume cannot be measured fundamentally by any 
process at all similar to those applicable to the magnitudes 
we have considered so far. Volume is measured (1) as a 
fundamental magnitude by means of incompressible fluids, 
or (2) as a derived magnitude by means of the lengths and 
angles characteristic of the surface. The second method 
depends upon numerical laws established by means of the 
first. In certain cases these laws can be related closely 
to other geometric laws by means of the following propo- 
sitions : — (1) Two complete surfaces with equal dimensions, 



28 Dr. Norman Campbell on the 

i. e. with equal lengths and equal angles between them, have 
equal volumes. (2) If two complete surfaces have each 
one part plane, and the boundary of the plane part of one 
can be brought into complete contiguity with the plane part 
of the other, then the complete surface which has dimensions 
equal to that of the complete surface so formed has a volume 
equal to the sum of the volumes of the original surfaces. 
These propositions could be used as definitions of equality 
and addition in a system of measurement, which would be 
independent of the measurement of length and angle (and 
therefore not derived), because it involves only equality, and 
not addition, of length and angle. But it is of limited scope 
and, in particular, would not permit the measurement of 
the volumes of curved surfaces. Since we do undoubtedly 
attribute a meaning to the volume of such surfaces, in a 
way that we do not to their area, measurement by incom- 
pressible fluids, which is not geometric, cannot be wholly 
avoided. But the propositions, which are those on which 
Euclid bases his treatment of volume, are actually used in 
modern practice, and are therefore regarded permissibly as 
laws of measurement. 

12. In deducing Euclid's propositions from the laws 
of measurement of these magnitudes, subsidiary laws are 
required, corresponding roughly to his postulates, expressed 
and implied. First, we need "existence theorems" corre- 
sponding to each of the definitions; for example, the 
definition of a plane surface justifies the conclusion that 
a straight edge can be placed contiguously to any two 
portions of such a part of a surface. Second, we need the 
assumption that we can make an object having a magnitude 
equal to that of any object presented to our notice. All 
these propositions are laws of measurement : the first group, 
because ail definitions in experimental science are nothing 
but existence theorems ; the second, because it is implied in 
the fact that we can make a standard series by which we can 
measure any magnitude. 

Euclid's three expressed postulates are all untrue. I 
cannot " draw a straight line " from this room to the next 
when the door is closed. Moreover his constructional 
propositions, closely connected with the postulates, are 
unsatisfactory because they are all directed to the drawing 
of scratches, rather than to the making of edges. The 
hypothetical experiments by means of which the deductions 
are effected are carried out much more easily with edges 



Elements of Geometry. 29 

than with scratches ; and if any of the propositions were 
doubted and put to the test of experiment, it would certainly 
be by means of edges ; the extension to scratches would be 
by means of the contiguity of edges with them. Euclid's 
methods here undoubtedly indicate that he is leaving, 
perhaps consciously, the realities of experimental science 
tor the pure ideas of mathematics. But he has made 
so little progress towards the new peak that, if he is to be 
restored to safety, it is far easier to drag him back to that 
which he lias never left completely than to guide him 
through the bog in which the two sciences are confused 
to the very distant goal. 

13. Only a few disconnected remarks will be offered 
here on the process of deducing the Euclidean propositions 
from the fundamental laws that have been sketched. Of 
course, we should employ the " application " (or contiguity) 
method of Prop. I. 4 wherever possible, instead of trying 
to avoid it ; for it is based directly on the fundamental 
notions. Again, we should not commit Euclid's error of 
supposing that strictly similar triangles can be brought 
into contiguity; we should apply the mirror image first 
to one triangle and then to the other. There would be 
no need to introduce area to prove Prop. I. 47. A Greek 
writer was forced to do so, because, not being familiar with 
the multiplication table, he could describe in no other w r ay 
the relation between a number and its product by 'itself. 
We should proceed from Prop. I. 34 to Book VI. and prove 
Prop. I. 47 by drawing the perpendicular from the right 
angle to the hypoteneuse and using the relations of similar 
triangles, treated by algebra. For nowadays, since w r e 
admit no incommensurable magnitudes, we can dispense 
altogether with Euclid's very beautiful and ingenious 
subtle! ies about ratios. A ratio in experimental science 
is nothing but a value taken from the multiplication table, 
which is established by the measurement of number, i. e. by 
counting. The laws of the measurement of number are 
involved in those of the measurement of every "continuous" 
magnitude. 

April 22, 1922. 



[ 30 ] 



III. On the Hotation of Slightly Elastic Bodies. By 
Dorothy Wrinch, D.Sc, Fellow of Girton College, 
Cambridge, and Member of Research Staff, University 
College, London * . 

THE change in dimensions of a slightly elastic body due 
to rotation is a question of some practical importance, 
and does not appear to have received any systematic treat- 
ment. In the theory of elasticity, the displacements of a 
point of the body are of course discussed and the displace- 
ments of the points of the boundary determine the increase 
of dimensions. But the problems of elasticity which are of 
interest mainly from the point of view of increase of dimen- 
sions, rather than of the distribution of stress in the material, 
can rarely be solved by the current methods or appear onlv 
as special cases of a general mode of analysis. Even the 
simple problem of a circular cylinder of finite length, rotating 
about its axis, has not yet admitted an exact solution, though 
an approximate solution, which becomes valid when the 
cylinder is of infinite length, has been given by Chree. 
When the cylinder has a finite length, the surface con- 
dition of zero traction over the curved surface is violated, 
and instead of this traction becoming zero at all points on 
the surface, only its average value over the surface is zero. 
The results for the case of an infinite cylindrical annulas 
do no.t appear to be on record, and they are interesting on 
account of their marked divergence from those which belong- 
to the complete disk. 

In the present paper we group together some of the 
simpler and more interesting solutions of problems of 
this type, including those of the infinite circular cylinder 
and the infinite cylindrical annulus. These specific pro- 
blems are solved to any degree of approximation and for a 
non- uniform distribution of density. The analysis is simpler 
than is usual, for it does not seem necessary to treat these 
comparatively simple problems as special cases of general 
theory, and it is desirable, at least in the interests of the 
engineer or physicist, that a fundamentally simpler treat- 
ment should be placed on record. It also seems possible 
that such solutions may be of interest with regard to 
scientific instruments of great precision, in which some 
portion of the apparatus is in rotation, or, on the larger 
scale, in problems of practical engineering. Although no 

* Communicated by the Author. 



On the Rotation of Slightly Elastic Bodies. 31 

novelty attaches to some of the earlier results, it seems 
desirable to include them. 

The simplest problem of! this nature is, of course, that of 
the thin circular hoop rotating about its centre. When such 
a hoop of radius a and density p is spun round its centre 
with constant angular velocity w the value of T, the tension 
per unit length in the Loop, is well known. For an element 
ds of its length has an acceleration aco 2 inwards, and the 
resultant of the tensions at its ends is Tds/a per unit area 
inwards. Hence the equation of motion is 

Tds/a = paco 2 , 

giving T = pa 2 co 2 . 

If, however, the hoop is slightly elastic, and X the value of 
Young's modulus for the material of which the hoop is 
made, and r the radius of the hoop when in motion, the 
equation of motion of the stretched element ds becomes 

T/V = ?'a> 2 . pair. 

Applying Hooke's law to the stretched element, we have, 

T = X(r-a)/a. 
Hence eliminating T, 

prato 2 = X(r — a) /a. 

In practice X is always large, and if we may neglect 1/X and 
higher powers of 1/X the appropriate value of r/a, which 
differs from unity by a quantity of order 1/X, is 1 + pa 2 co 2 /X. 
The value of the tension to the same order is pa 2 co 2 . 

The effect of a rotation is therefore to increase the radius a 
of the hoop to a(l +//,), where fj, = pa 2 co 2 l\, a number depending 
on the density, the elasticity, and the radius of the hoop, and 
on the rate at which it is rotating. 

As regards the practical order or magnitude of pa 2 (o 2 /X 
the extension per unit length, we may take a steel wire for 
which X is about 2*12 xlO 12 dynes per square centimetre, 
and p is about 7*5. In order that Hooke's law may hold, the 
extension per unit length must not exceed 10~ 3 , roughly 
speaking. If the velocity of a point on the rim is in the 
neighbourhood of 1*9 x 10 4 cm. per second — which is ap- 
proximately the case in a twenty-foot flywheel making two 
hundred and fifty revolutions a minute — we find that the 
extension per unit length is about 7*9 XlO -4 , which comes 
within the limits of applicability of Hookers law, and that 



32 Dr. Dorothy Wrinoh on the 

the actual increase in the radius is about a fifth of an inch. 
In this case the tension is about 1*6 x 10 9 . 

It is further evident that </ n \/p is the largest velocity 
if an extension of more than n per unit length is to be 
avoided. When the elastic limit for the material is known, 
this result can be used to give an upper limit to the velocity 
it is safe to use if risk of deformation of the hoop is to be 
avoided. 

We may now proceed to the problem of a thin rod rotating 
about one end with uniform angular velocity. 

Thin Mod Rotating about One End. 

Let a be the unstretched length of the rod, co the angular 
velocity of rotation about one end 0, p the density when it 
is unstretched, and X the value of Young's modulus for the 
material of which the rod is made. Let T be the tension 
in any section in the rod during the motion. Let the 
distances of the same particle at rest and in motion be 
x and x. The density of the moving element dx is 
podx^/dx and its acceleration towards is xco 2 . The equa- 
tion of motion of the element is therefore, 

ST = — p Bxq . xco 2 , 

where, by applying Hooke's law to the element originally of 
length dx and now of length dx, we have 

T = \(dx/dx -l). 

Hence, eliminating T, we obtain the equation, 

d 2 x/dx 2 = — p Q m 2 x / A . 

The solution must give the value x = a when x = a if a is 
the length of the rod when in motion. Accordingly it is 



x = asm(x V p (o 2 / '\) / sin (a Q ^p Q oo 2 j\). 

We may determine the value of a by means of the condition 
that the tension vanishes at the free end, which is given 
indifferently by x = a or x = a . Thus, 



« \/p G) 2 /\ = tan a \^p co 2 /\. 
The equation relating the two corresponding positions of a 



Rotation of Slightly Elastic Bodies. 33 

typical element when at rest and when in motion and the 
original length of the rod is therefore 



x \'p co' 2 /\ = sin (d? y/p w 2 /\) / cos (a vW*>7\)- 

Neglecting the cube and higher powers of l/\, we may 
replace this by the simpler form, 

as = a'o + 'i'opo(o 2 (3a 2 — tf 2 )/6\ 

to the order l/\. To the same order, 

T = p co 2 a 2 [1— x 2 la 2 ] . 

The greatest extension is ^p a) 2 a 2 /\, and this occurs at 
the end about which the bar is rotating. The tension is also 
greatest at this point and takes there the value p co 2 a 2 . 

As an example of the actual magnitudes of the quantities 
in practical cases we may take a twenty feet steel bar, which, 
when rotating about one end two hundred and fifty or three 
hundred times a minute, increases in length about a tenth of. 
an inch. 

dotation of an Infinite Elastic Circular Cylinder 
about its Axis. 

Passing now to a simple problem in three dimensions, we 
take the case of an infinite elastic cylinder of circular section 
rotating about its axis. We may consider one of the circular 
sections of the cylinder and use polar coordinates. At any 
point (r, 0) let r l\ and T 2 be the transverse and radial ten- 
sions per unit length, and T 3 the axial tension. We shall 
consider the motion of the element of volume which when 
at rest is bounded by the surfaces (z,z + 8z), (r, r + 6>), 
(0, + 80). By the symmetry of the cylinder, the element 
when in motion will continue to be bounded by the surfaces 
(0, + S0) : and since the cylinder is of infinite length, 
the element will continue to be bounded by the surfaces 
(z, z + hz). Let p represent the radial dimension, so that 
p — r is the radial extension at any point. Let a be radius 
of the cylinder and a its density, when at rest ; let &> be the 
angular velocity of the cylinder about its axis, and X and /x 
the elastic constants for the material of which the cylinder 
is made. 

The element of volume which we are considering is a 
parallelepiped of sides dp, pd0, and dz. The forces on our 
element of volume consist of (1) transverse tensions each 

Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. D 



34 Dr. Dorothy Wrinch on the 

o£ magnitude T±dpdz — and these are equivalent, in the 
usual way, to a radial force towards the centre of magnitude 

Tidpdsds/p, 

where ds~pd0; (2) of radial tensions 

T 2 ds dz and T 2 ds dz + dp d(T 2 ds dz)jdp, 
towards and away from the centre, which together give a 
£ ° rCe dpd0dzd( P T 2 )/dp 

away from the centre ; and (3) of longitudinal tensions each 

Fig.l. 




s p Ut 2 6s) 



T,t p/ .6s. 



of magnitude T s dp ds, in opposite directions. The resultant 
force then is simply 

dp dz ds (d(pT 2 )jdr — Tj) 

away from the axis, and perpendicular to the axis of the 
cylinder. The acceleration of the element is poo 2 towards 
the centre and its mass is ardrdO, since we may, of course, 
treat the density as constant over the element of volume. 
We therefore have the equation of motion, 

d(pT 2 )/dp — T 1 = — arpar dr/dp. 



d 
dr 



or 



Rotation of Slightly Elastic Bodies. 35 

By Young's law we can express T 1? T 2 , and T 3 in terms of 
the extensions of the cylinder, in the well-known equations, 

T^Xidp/dr-l) + (X + 2fjL)(p/r - 1), 

T, = (X + 2fiXdp/dr-l) + \(p/r-l) 9 

T 3 = \(dp/dr + p/r-2). 

Putting these values of T x and T 2 in the above equation we 
obtain the result 

,(X + 2,)g-l) + ^-l)] 

-SKS-- 1 )-^^- 1 )-^ 

Qa/r) rf^/Wr 8 + W) [( L V dp/dr)/(\ + 2/i) + {\p/r)/(X + !>)] 

[dp/dr-p/r] = -p<7G> 2 /(A, + 2/x). . (1) 

The value of //, or X, varies from about 8 x 10 8 grammes' 
weight per square centimetre for steel to about 4 X 10 8 
grammes' weight for copper. The corresponding densities 
are about 7 and 9 respectively. Terms involving 

0-/0+2/*) 

are therefore of a smaller order than those which involve 
coefficients of the form _ ,, _ \ 

2/*/(\ + 2,a). 

Let us write co 2 a — q{\~\-2pb). Then putting 

we can obtain a value for p to any order of approximation 
which is required. Neglecting, first, all terms involving q 2 . 
we have the equations for 77, 

p = r + qrj l ; dpjdr =l + q drjjdr ; d 2 p/dr 2 = q dPrjJdr* ; 
q [l + qvi/r] d?n 1 /dr*.+ llr[l + q(2/idf fl ldr)l(\ + 2p) 

+ i*>ViM/(\ + 2/*) + q (d Vl /dr- Vl /r)] = -qr(l + ? % /r) ; 
q dfyjdi* (1 + y i/x/r) + 1/r [1 + ? (2/t* d Vl [dr)j(\ + 2/,) 

+ frni/r)/[\ + 2/a)] 5 (d Vl /dr- Vl /r) +qr [1 + ^ r] = ; 
and since we are neglecting terms in y 2 , the equation for 77! is 

/•-' d 2 rji/dr 2 + r drj 1 /dr—rj 1 = — r 3 , 

D2 



36 Dr. Dorothy Wrinch on the 

giving a solution or* the form 

Vi^a^ + bjr-^r 2 

and b 1 = 0, since t] y is not infinite at the axis. The boundary 
conditions determine the constants a : and b x ; for the radial 
traction T x is given by 

T ] = (X +2 ,)(J-1) + X(^1) 

= q [{\ + 2 f L)d Vl /dr + \r ]l lr]\ 

and the radial traction must vanish over the two boundary 
surfaces. Thus 

(X + 2yLt) dffifdr | r =a +Xrj 1 /'r '< r =a =0. 

The constant a x is therefore determined by the equation, 

(\ + /x)a 1 = (2X + 3,a)a-78, 
giving 

T 1 = ? [2(X + /A )a 1 ~r 2 /4(2X+,>)] 

= 7(2X + 3/z)(a 2 -r 2 )/4r 2 
and 

7?1 = ^(a 1 r — ?' 3 /8) 

= ^[(2X + 3 / ,)(a 2 + & 2 )/ l X+/.)->- 2 ]. 

77i£ Effect of a Circular Hole in the Cylinder. 

If the cylinder at rest has two boundary surfaces r = a, 
r=b (b<a) the solution stands in the form, 

Vl = (ll r-^b l /r~ir\ 

and the conditions for zero radial traction on both the 
bounding surfaces yield 

(X + /x)a l — fjbbi/a 2 + (2X + 3/a) a 2 /S, 

(\ + fi)a l = /ib 1 /b' 2 i- (2\ + 3yu &-/8. 
These give 

a 1 = (2X-I- 3/x) {a 2 + & 2 )/M(X 1 /*), 

b\={2\ + 3fM)a 2 b 2 l${i. 
Thus 

T 1 =g[2(X + /A )a 1 -2 / ,/Vr 2 -(2X + 3^> 2 /4] 

= 7(2x + 3/x)(a 2 — ?•'-') (r 2 — b 2 )/-kr 2 . 



Rotation of Slightly Elastic Bodies. 37 

And the radial extension is 
p — r = <j \_a x r -\-b l ji — ? ,3 /8] 

o 

The maximum radial traction occurs where 

(« 2 -'- 2 )(>' 2 -i 2 )/>' 2 

is greatest, namely, at r 2 = ab. The greatest radial traction 
therefore occurs on the cylinder whose radius is the geo- 
metric mean of the radii of the bounding surfaces of the 
cylinder. Its value here is 

pco%2\ + -V)(a - by/MX + 2fi). 

The transverse tension is given by 

To = q [ [\ + 2/z) Vl jr + 2\ dih/dr] 

= 7 /4[(2X + 3/x)(a 2 + 6 2 ) + (2\ + 3^)a 2 6 2 /r 2 -(\ + 2^> 2 ] 

= (2X+3^)|[a 2 + 6 2 -a 2 & 2 /^-(\ + 2^)r 2 /(2\+3/ A )]. 

The maximum value of this tension occurs on the cylinder 
with radius r 2 given by 

r 2 = abi/{(2\ + 3fi)/(\ + 2fi)}, 

and there its value is 

7l2X ^ 3 ^[a 2 H-6 2 -2a/W(^ + L>)/(2X + 3^)]. 
Finally, the tension parallel to the length of the cylinder is 



_ fa 



•2 \ + 3/x 
X-\- a, 



<r 



+ 1 



2 )-2, 2 ]. 



The longitudinal tension T 3 is therefore greatest on the 
inner boundary. Here its value is 

q\ r(2\ + 3yu,Vr+ (fi-2\)h 



i\ n2\ + 3fju)a*+(fjL~2\ )(Sl 
4 L X + /x J' 



In the well-known case of a solid cylinder, the longitudinal 
tension has its maximum on the axis, and its value there is 

(2X-h3yLt)^ 2 X/4:(X+u). 



38 Dr. Dorothy Wrinch on the 

The radial and the transverse tensions have also their 
maximum value on the axis, where they take the same 
value, 

po) 2 (2\ + 3/x)/4a 2 (A + 2/x). 

The severest traction the cylinder is called upon to with- 
stand is therefore 

po) 2 (2A, + 3yu-)/4a 2 (\+2yu), 

and it occurs radially and transversely on the axis of the 
cylinder. 

In the case of the cylindrical annulus, the facts are entirely 
different. The longitudinal tension reaches its maximum 
value 

pco 2 X[{2Xi 3^a 2 /(\ + fi)-fjib 2 /{\ + fi)]mx^2fi) 

on the inner boundary. The transverse tension, in general, 
reaches its maximum value 

po>*(2\ + 3/*) [a 2 + b 2 -2ab V(X + 2m[2X + 3/*]/4(A, + 2<a), 

on the cylinder r = r 2 . And the radial tension reaches its 
maximum value 

f)co 2 (2\ + 3p){a-b) 2 mx + 2n), 

on the cylinder r=r 1 i As the ratio X//jl varies, the place at 
which the maximum tension occurs varies. If the ratio b/a 
is sufficiently large, i. e. if 

, /2\ + 3/x ,, . /a^+2/x 

b\f r t ->a, ba>\/— £-, 

V X+2[jl — ' ' ~ V 2X + 3/x 

the maximum transverse tension will occur on the outer 
rim ; otherwise it will occur within the cylinder. Thus a 
sufficiently thin annulus will have its maximum transverse 
tension on the outer boundary. For different values of the 
ratio X/fi, the maximum traction will occur transversely or 
longitudinally, for a cylinder for which the ratio a/b is given. 
And further, for a given ratio X/jjl, the maximum traction 
will occur transversely or longitudinally. 

Higher Approximations for Expansion of a Solid 
Infinite Cylinder. 

The first approximation for the radial extension was of the 
form, 

p — r-t-q {a x r — r*/8) = r + qn x . 




Rotation of Slightly Elastic Bodies. 39 

To obtain a second approximation we may put 

f/'-i + ^i/^fW''; 

and therefore 

dp/dr = 1 -f qdr}i/dr + qhirf^dr 

and neglect terms involving ^ 3 . Using, again, equation (1) 
we obtain, if terms in q* are again neglected, 

d 2 tj 2 /dr 2 + drji/rdr — 7J 2 /r 2 = — rj l9 

or r 2 d 2 rj 2 /dr 2 4- rdrj 2 /dr — r) 2 = — fl^r 8 + r°/8. 

The solution is of the form, 

r j2 = a. 2 r—a l r z /8 + r> 1 192. 

The radial tension T x is given by 

q [(X + 2p) d Vl /dr + X^/r] 4- q 2 [(X + 2/*) <fcfe/<*r + X V2 /r'] . 

The condition of zero radial traction on the boundary 
therefore yields 

(X + 2p) drj^jdr + Kvi/r + q[(X + 2p) d<n 2 Jdr + X7j 2 /r] = 

at r— a, or since the part independent of q already vanishes 
at r=a, 

(X + 2/jl) di) 2 ldr + Xrjo/r — 
at r = a. Hence 

T 1 = 7 (2\ + 3^)(a 2 -r0/4 

+ q 2 [(2\ 4- Sf^aja 2 - ?' 2 )/8 - (3X + 5p) (a 4 - r 4 )/S] , 
and since 

a 1 = (2\ + 3/A)a 2 /(X + /*) 
T 1 =q(2X+3fi)(a 2 ^r 2 )^ 

+ V 2 [(2X + 3/*)V(a 2 - r 2 )/8(X + //,)- (3X 4- 5/*)(a 4 -r 4 )/8],, 
and, finally. 

??2 /r = « 1 /[(2^-H3 A 6)a 2 /(X-r^)-^7 8 

- [(3X 4- 5/*K/0 + /*) - »'*] / 24 x 8 

_ 2X4-3/ , /2X 4- 3/, \ _ 1 / 3X + 5|* , _ A 

~ 8(xTm) V~X+>~ / 8.24 V X+/i j* 



40 Dr. Dorothy Wrinch on the 

Our second approximation is therefore, 

■2X + 3/JL 



XH-3, 

X + /JL 



' 2 r2\4-3/x 2 /2\+jV 2 2 \ 



24\ A, + ^ }]' 



Any higher degree of approximation can also be obtained. 
Our results stand in the form, 

and, in general, the form will be 

y s lr = d r 's + ia s r 2 . . . + s a s r 2s .... (2) 
We easily see that if (2) be the form for ij s /r, then ' 

and the condition of zero radial traction, requiring that 

(X + 2/a) drig+ijdr + \r) s+1 /r = 
on r=a, yields 

a g 2X + 3/A 9 2«« 3\4 5/x, 4 

uas+1 ~ 2 74TT7 a , + 176 ~Y+JT a ' * • 

6 .a s (s + l)\+(2s+l)fi 2s 

+ 2s-f-2.2s + 4 X + ^ 

so that 

W-2.4V ^ + y" J -4.-6V X + A* / 

+ 2s + 2.2s + 4V \+> / W 

When we have obtained a solution, 

p-r = q Vl + q2 V2 .. . +^ SJ 
it is therefore possible to obtain a solution to a higher degree 



Rotation of Slightly Elastic Bodies. 41 

of approximation by remarking the relation between the co- 
efficients in rj s jr and rj s +i/r and putting instead of 







as in 7j s l r, 

„ I ,— V / (» + l)X + (2n + ] y 2 „ + o ,,, + A 



Rotation of an Infinite Circular Cylinder of 
Non-uniform Density. 

We may next deal with the case of a cylinder in which 
the density is a function of the distance from the axis. 
Treating the case of the solid cylinder, we may put for the 
density of the cylinder when at rest, 

N 

*=J(r)=$a n r», 
o 

and N may haye any value from zero (when the density is 
uniform) to infinity — in which case the series '%a n r n must be 
convergent. 

The equation to be solved for a first approximation to the 
value of 7] is as before, 

d 2 7]/dr 2 + 1/r drjjdr — njr 2 = — aw'-rj (\ + 2/jl) . 

The solution is evidently, 

V =Ar- - ^ 2a M r»+7(( W + 3) 2 ~l). 

Tlie fact that there must be zero radial traction on the 
boundary surface r = a, yields the condition, 

T 1 =.(\+2/a) d V /dr + \r)/r = Q 
on r = a, giving 

2(X+/*)A 

S [(n+3)(X + 2f*)+X] a n a*+ 3 /(n+4)(n+2j 



X+2/*o 
=*r^ S [(fi+4)X+2(« + 3)/*] a„a^ 2 /(w + 4)(n f 2). 

K-t-lfi 



42 Mr. Gr. H. Henderson on the 

Consequently^ the value o£ T is 

r-^ r -2[(w + 3)(\ + 2 A t)+\] (a n+2 -r- +2 )a n /in+4,)(n^ 2) 
and 

2 

P ~ r ~\ + 2fi 

1ST 

V 

>-l 





XanK +2 (n + 4\4-2/i + 3yL6)/2(\ + ^)-r"+ 2 }/(v^ + 2)(// + 4). 



Higher approximations to the value of rj can be obtained 
by the method adopted in the case of uniform density. 



IV. The Straggling of a Particles by Matter. By Gr. H. 
Henderson, M.A., 1851 Exhibition Scholar of Dalhousie 
University, Halifax , JW.S* 

§ 1. Introductory . 

WHEN a parallel beam of « rays passes through matter, 
the particles gradually expend their energy in 
passing through the atoms of the matter, until all trace of 
the particles suddenly seems to vanish at the end of their 
range. In passing through the atoms some o£ the a particles 
lose more energy than others, so that at any point along 
their path some of the particles will be moving more slowly 
than others ; also their ranges will not all be the same. The 
a particles may be said to be straggled out, and hence the 
term straggling has been applied to "this phenomenon by 
Darwin. 

The theory of: the passage of matter by a rays has been 
developed on the basis of the nuclear structure of the atoms 
of the matter, and from this theory the amount of straggling 
to be expected has been deduced from probability, considera- 
tions. On the other hand, the straggling can be determined 
from experimental data in two ways. 

The first method makes use of ionization data. When the 
ionization due to a parallel beam of a, rays is measured at 
different points along the path of the rays, the well-known 
ionization curve is obtained. This curve is shown as the 

* Communicated by Prof. Sir E. Rutherford, F.R.S. 



Straggling of a Particles by Matter. 415 

lull curve o£ fig. 1, where ionization is plotted as ordinate 
and distance from the radioactive source as abscissa. Now 
it has been shown experimentally that the (average) velocity 
of the a particles at any point of their path is proportional 
to the cube root of their remaining range. Assuming that 
the ionization produced is proportional to the energy lost by 
the a particle at any point of its path, it can at once be 
shown that the ionization should be inversely proportional 
to the cube root of the remaining range. Such a theoretical 
ionization curve is shown as the dotted curve of fig. 1. It 

Fis-. 1. 




Rcinge 

will be seen to be in approximate agreement with experiment 
over the first portion of the path of the a particle, but as the 
maximum is approached this agreement fails. 

G-eiger * has suggested that the ionization curve observed 
for a beam of a rays should be different from that of a single 
a particle, owing to slight variations in the ranges of the 
latter, i. e. to straggling. The ionization curve, built up of 
a large number of theoretical curves grouped around one ot 
average range, will thus be modified considerably near the 
maximum where the ionization is changing rapidly Hence 
the shape of the ionization curve near the end of the range 
should give an indication of the amount of straggling. 
* Geiger, Proc. Roy. Soc. A. lxxxiii. p. 505 (1910). 



44 Mr. Gr. EL. Henderson on the 

Secondly, a more direct measure of the amount o£ 
straggling can be determined by counting the number of 
u particles at different points along the patli of a parallel beam. 

It is proposed in this paper to discuss the theoretical and 
experimental data on straggling, and it will be shown that 
the observed amount of straggling is much in excess of that 
allowed by theory. Further experimental evidence bearing 
on straggling will also be brought forward. 

§ 2. The Straggling in Air. 

It might be thought that the individual a particles are 
emitted with slightly different velocities, thus giving rise to 
straggling. It has been shown by Geiger (loc. cit.), however, 
that the a particles emitted from a thin layer of radioactive 
material do not differ by as much as -J per cent, in initial 
velocity. Thus the cause of the straggling must be looked 
for in the air itself. 

As the a particle passes through the air it gives up its 
energy to the electrons and nuclei of the air atoms, and it is 
occasionally deflected through a considerable angle by close 
encounters with the nuclei. Different a. particles will 
encounter different numbers and distributions of electrons 
and nuclei and accordingly are straggled out. The calcu- 
lation of the consequent probability variations in the ranges 
of the individual a. particles has been carried out by both 
Bohr * and Flamm |- They agree in showing that the 
nuclei produce practically no straggling. They also agree 
closely in the amount of straggling produced by the electrons. 
The straggling of various types of a rays in air, calculated 
by Flamm's method, is shown in the second column of 
Table I. The value tabulated is the distance, measured 
along the range, over which the number of particles in a 
parallel beam falls off from '92 to 'OS of the original number. 
This corresponds approximately to the method of measuring 
the straggling from the experimental curves. 

The ionization curves for three types of a. rays have 
recently been determined with some accuracy by the writer %. 
The full curve given in fig. 1 is a reproduction of the 
ionization curve found for RaC. It was shown that the 
ionization curve from C to B (fig. 1) could be represented 
very approximately by a straight line. The slope of this 
straight line furnishes information as to the magnitude of 

* Bohr, Phil. Mag. xxx. p. 581 (1915). 

+ Flarnm, Wien. Ber. II a. cxxii. p. 1393 (1913). 

X Henderson, Phil. Mag. xlii. p. 538 (1921). 



Straggling of a Particles by Matter. 



45 



the straggling. The. easiest way of considering the matter 
is to imagine the straight line produced in both directions 
till it meets the axes or! zero and maximum ionization at D 
and E. Then the projection of the line DE on the axis of 
zero ionization (or the reciprocal of its slope) is a direct 
measure of the straggling. The greater the straoro-lino- the 
greater will be the projection referred to, and as a first 
approximation the projection may be taken as proportional 
to the straggling. The values of the projections taken from 
the writer's curves are given in the third column of Table I. 
That for polonium has been determined from the curves 
given by Lawson *. 

The curves obtained by counting the number of a particles 
in a parallel beam at various points along the path show 
that this number remains constant till near the end of the 
range and then falls off rapidly to zero. Most of this falling 
off also approximately follows a straight line. The reciprocal 
slope or projection of this line is a more direct measure of 
the straggling than the corresponding projection of the 
ionization curve. Measurement of the straggling by means 
of counting experiments is, however, very slow, as large 
numbers of a particles must be counted. The values 
obtained from the most recent and reliable counting experi- 
ments are given in the fourth column of Table I. The 
result for polonium is taken from a scintillation curve given 
by Rothensteiner f ; that for RaC is from a curve obtained 
by Makower { by photographic counting of the a particles. 
All the results in the Table are in millimetres and refer to 
air at 0° C. and 760 mm. 



Table I. 



Type of Kays. 


Theoretical 
Straggling. 


Straggling from 
Ionization Curves. 


Straggling from 
Counting Expts. 




•88 
102 
144 
1-74 


3T 40 


; Thorium G x 


2-88 
2-83 
2-92 


4T 


: Thorium C 2 



This Table shows clearly that the observed values of the 

* Lawson, Wien. Her. II a. cxxiv. p. 637 (1,915). 

t Rothensteiner, Wien. Ber. II a. cxxv. p. ]237 (1915). 

% Makower, Phil. Mag. xxxii. p. 222 (1916). 



46 Mr. Gr. H. Henderson on the 

straggling are three or four times greater than those predicted 
by theory. Furthermore the calculated straggling increases 
steadily with increase of range, while that observed is 
constant within the limit of error. It should be pointed out 
that the projections given in the Table are measured as the 
small differences between two larger quantities, and hence 
are more difficult to determine with accuracy. The straggling 
deduced from the writer's experiments has a probable error 
of about 2 per cent., and it will be seen that the values for 
the three types of rays agree within this limit. 

It was shown by the writer (loc. cit.) that the effect of 
straggling due to electronic encounters would be a tailing 
off' of the ionization curve at the extreme end of the range. 
Making some simple assumptions it w r as shown that the 
calculated form of the end of the ionization curve agreed 
satisfactorily with the form of the curve observed between 
A and B (fig. 1). Thus the effect of the calculated straggling- 
was amply accounted for by AB, leaving the much greater 
straggling evidenced by the straight line portion BC quite 
unexplained. The curves obtained by counting experiments 
also lead to precisely the same conclusion. In view of the 
failure of theory to account for this large excess straggling- 
it is interesting to see what further information regarding it 
can be derived from experiment. 

It is remarkable that the straggling (as measured by the 
projections of the ionization and also of the counting curves) 
should be constant for a rays differing so widely in range as 
those given in Table I. This can only mean that the excess 
straggling takes place only in the last tw r o or three centi- 
metres of the range. From experiments with gold foils 
which will be discussed later, it appears probable that the 
straggling is confined to the last few millimetres of the 
range. 

Referring once more to fig. 1, it could not be expected 
that the straggling deduced from the ionization and the 
counting data would agree, for the following- reasons : — The 
ionization curve is regarded as being built up of simple 
curves of different ranges grouped about a common mean. 
The form of the simple curve is not accurately known ; the 
rule that the ionization is inversely proportional to the cube 
root of the remaining range can only be an approximation to 
a much more complicated law. As the shape of the simple 
curve cannot be taken into account, the projection of the 
ionization curve which is actually utilized can only give a 






. Straggling of a. Particles by Matter. 47 

rough indication of the absolute magnitude of the straggling. 
However, comparative values of the straggling under different 
conditions should be given fairly accurately by the method 
adopted. On the other hand, in the counting experiments 
the assumption is made that the zinc sulphide screens or 
photographic plates used have the same efficiency for a rays 
of low speeds as for those of high speed. This assumption 
is not altogether justifiable. 

§ 3. Straggling in Gases other than Air. 

The ionization curve in hydrogen was determined with 
the same apparatus already used for air. The gas was 
obtained from a cylinder of compressed hydrogen stated 
by the makers to be of more than 98 per cent, purity. 
Small impurities are unlikely to affect the straggling- 
mate rially. 

It was again found that a considerable portion of the eud 
of the ionization curve could be represented by a straight 
line. When the range of the a. particles was reduced so as to 
give the same range as in air, the projection of the straight 
line was 2*05 mm. with a probable error of 3 per cent. 

The straggling in air and hydrogen may be deduced from 
the ionization curves given by other observers. The results 
agree in every case within the limits of error, although the 
conditions for accuracy were less favourable than in the 
present experiments. The value 2'0 mm. for polonium in 
hydrogen may be obtained from some ionization curves given 
by Taylor *. From the results of Lawson (loc. cit.) for 
polonium the straggling was determined as 3"1 mm. in air 
and 2'2 mm. in hydrogen. 

The straggling in oxygen has also been deduced from 
experiments made in that gas with the present apparatus, 
using ThC. The value found was 3*36 mm., when the range 
was increased to the same value as in air. 

The straggling in several other gases may be deduced 
from the ionization curves given by Taylor (loc. cit.), although 
the error involved is probably of the order of 10 per cent. 
The collected results of stragglingin gases are given in fig. 2, 
which shows the straggling plotted against molecular stopping 
power. The values plotted for air, hydrogen, and oxygen 
are from the writer's results; the remainder are taken from 
es. 

* Taylor, Phil. Mag. xxi. p. 571 (1911). 



48 



Mr. G. H. Henderson on the 



Fig. 2 can only be considered to give an approximate 
idea of the .facts, as the points are not well distributed and 
some may be seriously in error. It would seem, however, 
that the straggling increases very slowly as the stopping 
power of the gas is increased. It is unfortunate that the 
dearth of suitable gases of high stopping power makes the 
checking of this point difficult. 

Fig. 2. 



. 4 

£ 
E 
c 3 

O") 

c 
o-» 2 

<3 



if) t 



so, £ H * a 




O 2 J*-*—"" 

Air *'" cs d 


<^ H ,o° 


. ._!_. _JL_ ... .1- 





1 a 

Molecular S+o 



pp.ng 



3 

Power. 



§ 4. T7i£ Straggling due to Solids. 

The great difficulty which at once arises in determining 
the straggling due to solids is the uneven thickness of the 
solid foils used, the effect of which may completely mask 
the true straggling looked for. An attempt to avoid this 
difficulty was made by using a large number of the thinnest 
beaten foils of the solid obtainable ; with gold, for example, 
as many as 128 thicknesses were used. Composite sheets of 
gold and other metals were placed immediately over the 
radioactive source (ThC) and the ionization curves deter- 
mined in air with the same apparatus as before. Although 
a rough calculation seemed to show that the irregularities in 
the individual foils would be smoothed out enough to avoid 
masking any true increase in straggling, this result was not 
borne out by experiment. It was finally concluded that the 
increase in straggling observed was mainly, if not entirely, 
due to unevenness of the foils, and hence need not be gone 
into in detail here. In mica the increase in straggling was 
much the smallest, as was indeed to be expected. 

Fortunately, experiments on the straggling produced by 




Straggling of a. Particles by Matter. 



49 



solid foils for low velocities of the a particles gave results 
which were not masked by irregularities of the foils. In 
these experiments the foils were placed 3 mm. from the 
middle of the ionization chamber (itself 1 mm. dee])) in air 
at a pressure of roughly 17 cm. Reduced to air at normal 
pressure the distance from foil to centre of ionization chamber 
was therefore about "7 mm. Most of the foils used were 
made up of a few T thicknesses of gold leaf. The air equivalent 
of a single sheet was about *I5 mm. when placed directly 
over the source ; when placed near the ionization chamber 
the air equivalent was about '28 mm. The straggling of 
the a particles after passing through these foils was deter- 
mined in the same manner as before from the ionization 
curve in air. 

The results are given in Table II. The straggling is in 
millimetres, and the probable error is about 2 per cent. The 
third column shows the straggling observed when the foils 
are placed directly over the source, the steady increase with 
increasing number of leaves being mainly due to unevenness 
of foils. The fourth column shows the increase in straggling 
at low velocity over that at high. This .increased straggling 
is real and almost independent of the unevenness of the 
foils. Results with aluminium (1*0 mm. air equivalent) 
and mica (S'6 mm. air eq.) are also included in the Table, 
but with these foils the change observed is scarcely more 
than the experimental error. 

Table II. 



No. of Leaves ( ^™ 8 loHifa«on Stra ^ in S Foil 
111 FolL Chamber. near Source ' 


Difference. 


2-88 

1 3-30 

4 ■ 3-65 

12 400 


2-88 
2-92 
301 
3-28 
3-01 
306 


•38 
•64 
•72 
•07 
•10 


Aluminium ' 308 

Mica 3-16 



These results show quite clearly the rapid increase of 
straggling near the end of the range. One gold leaf nearest 
the ionization chamber causes nearly twice as much straggling 
as three leaves immediately behind it. Eight more leaves 

Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. E 



50 



Mr. G. H. Henderson on the 



placed behind these four again only slightly increases the 
straggling. 

The same result was also demonstrated in a slightly 
different manner. A foil made up of the four gold leaves 
already used was placed at different distances from the 
ionization chamber and the straggling determined from the 
ionization curves as before. The results are shown in 
Table III. The distances given in the first column are not 
the actual distances from the foil to the centre of the 
ionization chamber, but are reduced to correspond with an 
ionization vessel containing air at atmospheric pressure. 

Table III. 



Distance from 
Ionization Chamber. 


Straggling. 


•7 mm. 
4-7 „ 
12-9 „ 


3'65 mm. 
3-26 „ 
2'95 „ 



Here again it can be seen that the straggling increases 
rapidly near the end of the range. 

It may also be noted that the straggling (2*95 mm.) when 
the foil was 12*9 mm. from the ionization chamber is less 
than that (3 01 mm.) obtained when the foil was placed 
directly over the source. The difference is less than the 
possible error, but such a change is to be expected. For 
the straggling due to unevenness of the foil should be less 
for low velocities of the a particles on account of the decrease 
in air equivalent of the foil. For the same reason the differ- 
ences given in the fourth column of Table II. are probably 
slightly too small. 

From the straggling observed with different types of 
ol particles it was pointed out in § 2 that the straggling must 
occur within the last two or three centimetres of the range ; 
from the results with gold foils it seems that the straggling 
must be confined largely to the last few millimetres of the 
range. 

The increase of straggling due to gold foils placed near 
the end of the range, though clearly marked, is small. 
This is quite in accordance with the view expressed in § 3 
that the straggling does not increase rapidly with increase 
o£ stopping power of the substance causing the straggling. 
It must be remembered that the increase - observed is the 



Straggling of u Particles by Matter, 51 

increase over the straggling which would be produced by a 
layer of air equivalent to the gold foil used. The ionization 
must be measured in a gas such as air at a reasonable 
pressure, and hence we have the complication of the straggling 
due to the air between the solid and the ionization chamber, 
and even in the chamber itself. This sets a limitation to 
the amount of information to be obtained From ionization 
data. Accordingly the ionization experiments were not 
carried beyond the stage described, but it is hoped to push 
further the attack on the problem by more suitable methods. 
An isolated experiment with iron foils may be referred to 
before closing this section. A sheet of iron of about 4 cm. 
air equivalent was placed directly over the source and the 
ionization curve measured when the iron was magnetized 
parallel with and perpendicular to the direction of travel of 
the a particles. The purpose of this experiment was to see 
if there was any change in the straggling due possibly to 
rearrangement or change of orientation of the electrons 
in the iron, a point which has been discussed by Flamm *. 
Alternate readings of the ionization with parallel and per- 
pendicular fields of about 100 gauss were made at various 
points along the range. No appreciable difference could be 
detected- It should be added that the iron was very uneven 
in thickness, and the consequent; straggling was so large 
(about 20 mm.) that a small change in straggling might 
well have been masked. 

§ 5. Summary and Conclusion. 

It has been shown in this paper that the straggling of 
« particles, as deduced from both ionization and counting 
experiments, is several times greater than that deduced from 
theory based on our present views of the mechanism involved 
in the passage of a particles through matter. It has been 
shown in a previous number of this Magazine that the effect 
of the calculated straggling can be adequately accounted for 
by the tailing off of the ionization curve at the extreme end 
of the range. The large additional straggling observed 
behaves quite differently. Evidence has been given in this 
paper to show that it increases very slowly with increase of 
molecular stopping power, and furthermore, that it all takes 
place within the last few millimetres of the range. Here we 
seem to be confronted with a behaviour of the a part cle 
which present theory is unable to explain. 

* Flamm, Wien. Ber. II a. cxxiv. p. 597 (1915). 
E 2 



52 Dr. G. Green on Fluid Motion 

Evidence obtained from the Wilson photographs also leads 
to the same idea, for it has been shown by Shimizu * that 
the observed number of ray tracks which break up into two 
branches near the end of the range is much greater than the 
number deduced from probability considerations based on 
our present theory of atomic structure. 

It is noteworthy that this anomalous behaviour of the 
a particle occurs at low velocities, where practically no 
investigation of the scattering of a particles has been carried 
out on account of the experimental difficulties of dealing 
with slow ol particles. It is at higher velocities, where the 
theory of scattering put forward by Sir Ernest Rutherford 
has been so fully verified by experiment, that the most of 
the theoretical straggling takes place, and this straggling- 
has apparently been accounted for. 

In conclusion I wish to express my best thands to Professor 
Sir Ernest Rutherford for his kind interest and advice. 
I also wish to thank Mr. Crowe for the preparation of the 
radioactive sources. 



Y. On Fluid Motion relative to a Rotating Earth. By 
George Green, F>.Sc, Lecturer in Natmal Philosophy in 
the University of Glasgow f. 

THE subject of this paper is at present one of consider- 
able interest to meteorologists. Papers by the late 
Dr. Aitken and also by the late Lord Rayleigh on the 
dynamics of cyclones and anticyclones have been followed 
by more recent papers by Dr. Jeffreys, Sir Napier Shaw, 
and others. Very few actual solutions of the equations 
defining atmospheric motions have been obtained. In the 
late Lord Rayleigh's paper % attention is drawn to certain 
general hvdrodynamical principles relating to the properties 
of rotating fluid which can be applied to " assist our judg- 
ment when an exact analysis seems impracticable." The 
importance of the theorem regarding the circulation of the 
fluid in any closed circuit is clearly explained in its applica- 
tion to any actual fluid motion. In applying this theorem 
to fluid motion in the atmosphere, however, we must bear in 
mind that the motions with which w r e are concerned are not 
the actual motions of the particles in space but their motions 
relative to the Earth itself at each point of observation. 

* Shimizu, Proc. Roy. Soc. xcix. p. 432 (1921). 
f Communicated by the Author. 
X Sc. Papers, vol. vi. p. 447. 



relative to a Rotating Earth, 53 

One object o£ the present paper is to investigate the con- 
ditions under which the circulation theorem may be applied 
to atmospheric motions relative to the Earth's surface ; or 
more generally to motions relative to any three rectangular 
axes which are themselves rotating about each other, with a 
fixed origin. In the later part of the paper one or two 
additional cases of motion of the atmosphere are discussed 
and the system of isobars corresponding to each motion 
determined. 

In view of the problems to be considered, we shall begin 
by specifying the system of rotating axes most convenient 
in dealing with fluid motion in the neighbourhood of any 
point of reference on the Earth's surface. The axis OZ 
is drawn upwards along the apparent vertical at 0, and 
line OZ continued downwards meets the axis of the Earth 
at a point 0' which is taken as origin of coordinates. Then 
axes O'X and O'Y are drawn parallel to horizontal lines 
through the reference point in directions due East and 
due North respectively. In the most general case to be 
considered the reference point may be in motion relative 
to the Earth's surface, and this involves also a motion of the 
origin 0' if point moves either North or South. But 
the motion of 0' corresponding to any moderate motion 
of is very small, and for our present purpose we may 
regard the origin 0' as a fixed point, very near to the centre 
of the Earth. We shall denote by (.r, y, z') the coordinates 
of any point referred to origin 0', and by (#, ?/, z) the co- 
ordinates of the same point referred to parallel axes through 
0. This makes z r = z + H, where R represents approximately 
the radius of the Earth. The components of the velocity of 
any particle relative to the axes at any instant are repre- 
sented by u, r, w, and the angular velocities of the axes 
themselves, that is, of each two axes about the third, are 
represented by ft> z , ay^ a) z , respectively. We shall introduce 
the particular values of ao x , w y , co z corresponding to a reference 
point fixed in position on the Earth, or moving relative 
to the Earth, when we come to deal with special problems. 
Referred to the above system of axes, the equations of motion 
of any fluid particle take the form : — 

Dz < , a - <* V 1 "dp ( i n 






54 Dr. G. Green on Fluid Motion 

In these equations, 1? 2 , Z represent the terras depending 
on the rotation of the axes themselves, being given by 
equations of the type 

#i = — 2o) z v + 2<Djiv — w : y + Wy.z 1 + co x (o y i/ -f ov^s' — (co/ + &)/) 0. 

... (4) 

The function V(#, y, *) represents the gravitational potential 
function. We have also 

D 3 ^ 9 j. „ ^ j. * «•> 

D7 = 5t + l( ^ + ^ + "5i- • • • (5) 

The equation of continuity of the fluid is then 

W + K^ + ^ + 9l)=°- • • • (b) 

In applying these equations we treat the atmosphere as a 
perfect gas in which viscosity may be neglected. 



Circulation Theorem for Relative Motion. 

Consider now the theorem relating to the relative circula- 
tion. We have 

j^ (u dx + vdy + w dz) = y- dx + jydy + -^ dz + d (i q 2 ), 

• • • (?) 

where q 2 = u 2 + v 2 + w 2 , the square of the resultant relative 
velocity. By means of equations (1), (2), (3), the above 
equation may be rewritten in the form : 

yr- (u dx + v dy + w dz) 

= -(0 L dx + 6 2 dy + s dz)- C ^-dY + d(iq 2 ). (8) 

We can now integrate each term of this equation along any 
curve within the fluid from any point A to any point B. 
This integration gives the result, 

Dl'B rs C dp 

n I (udx + vdy + wdz) =—] (#i dx -\-0 2 dy + O s dz) — 1 - J - 

-V B + V A + i?B 2 --kA 2 ; (9) 

and, if the integrations are applied to a closed curve 



relative to a Rotating Earth. 55 

beginning and ending at the point A, we obtain 

D C C 

jj-l (uds+vdy+wdz)= — \ (d l dx + e 2 dy+6 z dz), (10) 

v s * s 

where the suffix S indicates that the integration is to be 
taken along a definite curve S. We have assumed in 
obtaining (10) from (9) that V is a single valued function 
of («r, i/, z), and that p is a f miction of p. It now appears 
that the rate of change of the relative circulation in any 
closed circuit which consists of the same fluid particles at 
all times is not zero unless, in addition to the above con- 
ditions, we have 

B01_d03 ^2_Bfl ^3_^2 nn 

When these conditions are not fulfilled, the relative vorticity 
does not move with the fluid itself, and if a velocity potential 
exists for a certain portion of fluid at a given instant, a 
velocity potential will not exist for that portion of fluid at 
a later instant. 

The first case of importance of the above conditions in 
relation to problems relating to the atmosphere is that 
in which the angular velocities co x , o> y , co z of the axes are 
constants. In this case, the conditions given above take the 
form 

1 * 1 o 1 £ "bit "dv "dw n9 N 

(D r Wy co, o<v oy 0% 



where 8 represents an operator defined by 

'-(•4, + +fr-h)- • • • (13) 

These equations have a solution of the form 



u v ic 



,-^= «=/(•*+-#+-/)> ■ ' (14) 

where /denotes any arbitrary function. If we draw an axis 
to coincide with the axis defined by the resultant of the 
three component rotations co X) co y , co z , then [co x x + co y y + co z z) 
is equal to URcos^), where fl is the resultant of (a> x , a> y , (*> z ) 
and R is the line joining the origin to the point a, y, z. 
That is, u, v, w are functions of p the perpendicular from 
the origin to a plane through the point (<r, ?/, z) perpen- 
dicular to the axis of the resultant rotation. 



56 Dr. Gr. Green on Fluid Motion 

When the fluid is incompressible, and when a compressible 
fluid is moving m such a way that ^ — h ^ — \- %- is zero, 

B# oy d~ 
a solution of a different type obtains. The solution in this 
case may be written in the form 



u 



I 



=/i{(V-^), (a z x-co x z)}, 

v=j 2 {{a) 1J x-G) x y), (ay z x-co x z)}A. . . (15) 

w =/ 3 { (aye - <o x y) , {co z x - m x z) }, ) 

where f 1} f 2 , f% are arbitrary functions subject only to the 

condition — + ~- + ^- =0. This solution includes as a 
ooc dy oz 

particular case any motion of rotation of the atmosphere as 
a solid about the axis of the Earth. 

The solutions which we have above obtained make it clear 
that the fluid motions relative to rotating axes in which the 
relative circulation moves with the fluid belong to a very 
restricted type. A relative motion, for instance, similar to 
that taking place in a free vortex, does not fulfil the con- 
ditions required for permanence of the velocity potential, 
and therefore no steady motion of this type could take place 
in the atmosphere — as has been assumed to be the case. 

The conditions which we have found to be necessary for 
the validity of the circulation theorem when the fluid motion 
is relative to rotating axes, may be obtained in a manner 
different from that employed above. Taking H, H, Z to 
denote the components of angular velocity of a fluid ele- 
ment, and U, V, W to denote components of linear velocity 
of the element, each referred to fixed axes which coincide at 
instant t with the instantaneous positions of the moving axes, 
we may derive the conditions from the equations employed 
by von Helmholtz in his papers on vortex motion : — 

with two other similar equations. With £, rj, £ to represent 
the components of relative angular velocity of an element of 
fluid, referred to the moving axes, we have, 

JJ = u — (o z y + cOyZ; Y = v — co x z + a) z x; W = to — w y x + w x y ; 
and Dt l)t ~ a ' v W ^' 



relative to a Rotating Earth. 57 

By means of these relations we can readily transform (16) 
and obtain the corresponding equations for the rates of 
change of the circulation components of an element of fluid 
referred to the rotating axes ; in this way we find 

Df <-B« . B« . V B" f./'d" , Bw . "dw\ 

d# B// Z B^ \B.i' By Bs/ 

with the corresponding equations in rj and f. Now the 
hydrodynamical theorem that relates to the permanence 
of a velocity potential for the motion of a given portion of 
fluid and the theorem of the permanence of the circulation 
of an element of fluid depend on equations (16). The 
equations which we have obtained for the relative circula- 
tions reduce to these equations exactly when the conditions 
expressed in (12) are fulfilled ; and these conditions must 
accordingly be fulfilled in order that the theorems referred 
to may apply to the relative motion, in the same way as they 
apply to the actual motion. 

Particular Cases of Motion Relative to the Earth, 

We shall now discuss one or two particular cases of fluid 
motion relative to the Earth, and we shall, to begin with, 
take the reference point O as a fixed point on the surface of 
the Earth at latitude (j> degrees North. The angular velocity 
components w x , a> v , co z have then the values 0, fl cos <£, 
12 sin </>, respectively, where fl represents the angular 
velocity of rotation of the Earth about its axis. If we now 
let zr represent the perpendicular distance from any point 
#, y 9 z to the axis of the Earth, we can write the equations 
of motion (1), (2), (3), in the form 

where V'=Y— JDV. V is, in fact, the potential function 



dV gx 


3V gy 


av 


a« ~ r ' 


ay R' 


S* 



58 Dr. G. Green on Fluid Motion 

TsY' 

corresponding to apparent gravity, so that — ^ — at point 

(# = 0, y = 0, £ = 0) is the value of —g at the reference 
point 0. In applying the above equations to motion of the 
atmosphere, we may take 

av av av t . 

in the immediate neighbourhood of in the region within 
which the value of apparent gravity may be regarded as 
constant in direction and amount. If we neglect a change 
o£ direction of one degree in g, our equations (18) to (21) 
would then represent conditions of motion within a radius 
of about seventy miles from point O. In order to render our 
equations suitable to represent circulations of air of diameter 
exceeding, say, 150 miles, we might employ the approximate 
values 

, . . (22) 

wherein we neglect the variation of g with height. 

If we exclude certain cases of motion relating specially to 
the tides, very few solutions of the above equations have been 
recorded. In order to make ourselves familiar with the 
types of fluid motion possible in the atmosphere it is of 
interest to examine all solutions which can be obtained 
having a bearing on meteorological problems. We, accord- 
ingly, take first the steady rotational motions of incom- 
pressible fluid under the force of gravity alone. 

We may take the boundary condition w = Q to apply at 
the surface of the Earth. A simple rotation of all the fluid 
about a vertical axis through 0, with a uniform angular 
velocity o>, would be represented by 

u=—(oy; v=+(Die; io = 0. . . (23) 

With these values in equations (18) to (20), it would be 
impossible to satisfy (19) and (20) simultaneously ; but 
the motion represented by 

u= — co(y — /3z) ; v = cox\ io = 0, . . (24) 

fulfils all the conditions contained in our equations, pro- 
vided (3 is given by 

o_ 212 cos j> m . 



relative to a Rotating Earth. 59 

and the pressure p is given by 

? = - V + i(a> 2 + 2a>n sin (/>) {.i< 2 + <j/-/3c) 2 } + & . (26) 

» r 

The approximate value of V' is +g[ 9T / + - ) , and p is 

the pressure at the reference point 0. In the motion indi- 
cated above each particle of fluid moves in a horizontal 
circle whose centre lies in the line (y=/3z, a? = 0). This line 
lies in the meridian plane through point and is inclined to 
the vertical at at an angle 6 towards the North, where 
tan<9 = /3. When a) is very small in relation to H this in- 
clined axis is almost parallel to the axis of the Earth ; and 
when co is large this axis comes almost to coincidence with 
the apparent vertical at 0. With co very large the motion 
described above corresponds very closely with a uniform 
rotation of the fluid about a vertical axis, as in the case of a 
simple forced vortex. 

Another case of motion of incompressible fluid of interest 
in the same connexion is that represented by 

.,, u=—coy; v = cox; w = 2X2 cos 6 . a?, . (27) 

with T 

J - = -V' + i (co 2 + 2vn sin (f>){% 2 +y 2 ) +2W cos 2 cb . x 2 + & , 

. . . (28) 

as the equation showing the distribution of pressure. This 
motion differs from that first discussed in not being exactly 
horizontal. The plane of motion of each particle of fluid 
passes through the line OX, and is inclined to the horizontal 
plane XOY at an angle 6 given by tan = 2Q cos <p/co. 
When the angular velocity of rotation co is very large com- 
pared with flcos</), the plane of motion of each particle is 
practically horizontal, and the motion then corresponds very 
closely with that of simple rotation of all the fluid as a solid 
about, a vertical axis. When co becomes small, on the other 
hand, the inclination of the plane of motion of each particle 
of fluid to the horizontal increases. The two motions, repre- 
sented by (24) and (27) respectively, are almost identical 
when ft) is very large, and they differ entirely when co is 
very small. It would be interesting to investigate the 
manner in which a fluid, such as water, subsides to rest 
from an initial condition of steady rotation about a vertical 
axis. The solution represented by (24), (26), would appear 
to be the exact solution for steady rotation of water in small 
scale experiments. 



60 Dr. Gr. Green on Fluid Motion 

The motions considered above are motions of any incom- 
pressible fluid and do not indicate, except as approximations, 
conditions of motion possible in the atmosphere, One or 
two solutions of a similar type can be obtained which refer 
to incompressible fluid and accordingly represent motions 
possible in the atmosphere. Consider now the motion repre- 
sented by 

«=-©(y-^); 5 v=0; m? = 0. . . (29) 

The equations of motion of any element of fluid in this 
case are 

°=- 9 i- k h logp ' ■ (30) 

-20 sin </> . oo(y-!3z) = ~ || -k^-logp, . (31) 

2OooB*«(y-^)=-^-^lo g/0 , . (32) 

and these equations are satisfied provided the pressure 
system throughout the fluid is that indicated by 

ilogp = n«a*. a>(y-/3zy- 9 (^f + «) + Mogp , (33) 

where /3 = cot.$, and p is the density of the fluid at the 
reference point O. The continuity equation is also satisfied 
provided we can neglect the term (go)/R),v(y — ftz). This 
condition limits considerably the extent of the region 
around to which our solution is applicable, as stated 
earlier. Within the region to which the above applies the 
isobars at the surface of the Earth run East and West, 
being determined by 

h log p = 12 sin </> . coy 2 -f k log p Q , . * . (34) 

which indicates a system symmetrical on the two sides of 
the East and West line drawn through reference point 0. 
In this case the isobars become closer as we proceed North 
or South from point 0. They are also parallel to the lines 
of flow of the fluid. 

The coefficient ^ has the value 1*5 x 10~ 6 with the foot 

and the second as units; while 2D sin (j> has the value 
1*03 x 10 ~ 4 at latitude 45°. Certain cases of interest arise 

in which the terms containing ~ may be neglected. For 



relative to a Rotating Earth, 61 

example, we may take the case of a uniform east or west 
wind over a considerable region. In this case 

u = c, r = 0, w = 0, . . . . (35) 

and the pressure distribution consistent with this motion is 
represented by 

k log /3 = 2 Or (cos </> z — sin (j>y) — gz + k\ogp , . (36) 

where p Q refers to the density of air at the reference point 0. 
The isobars at the Earth's surface are in this case a uniform 
system running due East and West. 
A similar case is that given by 

u=0, v = cx + d, 20 = 0, . . . (37) 

which represents a wind towards the North, while the corre- 
sponding pressure distribution is that represented by 

k log p = O sin (/> (ex 2 + 2dx) —gz + h log p . . (38) 

The isobars are again a system of straight lines, but runnino- 
north and south, and uniformly spaced when c = 0. 
In a similar manner we find that 

u = c 1 , v = c 2 , iu = 0, . . . . (39) 

corresponds to a system of straight isobars represented by 

k \ogp = 2Q sin cf) {c^x — c^y) + 2Q cos<f> .c^—gz + k log/j . (40) 

The isobars are again lines of flow of the air, as in each case 
considered above. 

The case of motion corresponding most closely to a 
cyclonic or anticyclonic circulation is that discussed in an 
earlier paper *, represented by 

it=—<o(y-(3z); v = co l i-; iv = 0. . . (41) 
In this case 

Q _ 2fl cos <j) 

^- w + 2min^,' < 42 ) 

and the pressure distribution is that represented by 

^log^ = i(a) 2 4-2a>f2sin</,){.r 2 -r( i/ -/3c) 2 }--^- f ^log /0o . 

. . . (43) 
The term fr^- must be small in order that the continuity 
equation may be fulfilled. 

* Phil. Mag. vol. xli. April 1921 ; vol. xlii. July 1921. 



62 Prof. G. N. Antonoff <m the 

In each case considered we have only to replace O by an 
increased value fl' to obtain a motion for which the system 
of isobars travels eastward at a uniform speed (CI' >Q). 

Each of: the above solutions has been given to apply to an 
isothermal atmosphere, and in every case considered the 
fluid moves so that no element of fluid undergoes change of 

density. That is, ~ — h s: — H ^— =0 at each point. Provided 
"* 3# dy d* r 

this condition is fulfilled, any solution obtained for motion of 
an atmosphere, all at one temperature, can readily be trans- 
formed to suit an atmosphere in convective equilibrium 
(p = Jepy), or one in which pressure is any given function of 
density. Thus, taking the motion represented by (41) above 
in an atmosphere in convective equilibrium, we have merely 

to replace log/o and log/o in (43) by — —. ,P y ~ l and - po 7 " 1 

respectively, all other conditions being unchanged. 



VI. The Breaking Stress of Crystals of Rock- Salt. 
By Prof. G. N. Antonoff, Z>.Sc.(Manch.) *. 

IN a paper published in Phil. Mag. vol. xxxvi. Nov. 1918, 
I have developed a theory of surface tension under the 
assumption that the attraction of molecules is due to electrical 
or magnetic forces, or both. Instead of assuming a uniform 
field round the molecules as it is generally accepted acccording 
to Laplace, I accepted the view that the molecules act as 
electrical doublets, and from the theory of potential I 
deduced that the attraction between them must be inversely 
proportional to the 4th power of the distance, provided the 
distance between the doublels is large compared with their 
respective lengths. 

It was shown that the attraction between the doublets can 
be represented by an expression of the type 

kP 

where k is a constant, I the length of the doublet, and d the 
distance between them. In these calculations the magnetic 
forces were disregarded altogether, as the law of attraction 
between small magnets would be just the same, so that they 
could only have an effect on the value of k. 

* Communicated hy Dr. J. W. Nicholson, F.R.S. 



/ d> 



Thus 



P 9 / S l-V 1/E 



Breaking Stress of Crystals of Rock- Salt. 63 

For the surface tension, the expression was given as 

d 

or assuming ~r=P where p is the number of molecules per 
unit volume, the expression for the surface tension a becomes 

u = kl 2 p^; 

instead of p we may put l 2 , where 6\ is the density of 

the liquid, B 2 that of the saturated vapour, and M the 
molecular weight. 

-=«■(¥•)" (i > 

It was also shown that the internal pressure P can be 
calculated by the formula 

P = 2«p 1 / 3 (2) 

) (3) 

In other words, the intrinsic pressure can be calculated 
from the surface tension if the molecular weight of the 
liquid is known. Thus for normal or non-associated liquids 
it should be possible to calculate the normal pressure from 
the value of the surface tension. 

It should be pointed out that the assumption made in the 
above theory, that the length of the doublet is small compared 
with the intramolecular distance, is not necessarily the right 
one. 

In the expression (2) this law is, however, eliminated, and 
the same expression is obtained for any law of molecular 
attraction. The figures for P obtained from the above 
expression agree with those from indirect evidence. How- 
ever, experimentally it is not possible to determine P 
directly, owing to the mobility of the particles of liquids 
which always adjust themselves so that the molecular 
pressure is inappreciable. 

It is not so in the solid or crystalline state, in which the 
particles have a definite orientation, and where the internal 
pressure can be determined by a direct experiment. It is 
sufficient to apply to a crystalline body such a weight as 
would overcome the attraction of the molecular forces and 
cause the disruption of the body. The force applied is not 



64 Prof. G-. N. Antonoff on the 

necessarily the same in all directions, and it is therefore 
necessary to specify the direction in which it is to be applied. 

The question arises now whether it is possible to calculate 
the surface tension of a solid body. 

For the solid state there is no direct method of determining 
the surface tension, all methods used for liquids being 
inapplicable in this case. Some attempts were made to 
estimate the surface tension of solids from indirect evidence. 
For example, Ostwald * and Hulett f calculated the surface 
tension of some calcium and barium salts on a basis of a 
certain theory from the solubility data. The figure given 
for the latter is about 4000 dynes per cm. From the point of 
view of our theory, it seems possible to calculate the surface 
tension by the use of formula (2) by determining experi- 
mentally the internal pressure per square cm. of the cross- 
section, if the molecular structure of the substance in the 
crystalline state is known. 

At the present time the X-ray analysis throws a light on 
the above question. 

• For example, according to W. H. and W. L. Bragg, the 
crystal of rock-salt consists of charged ions situated at regular 
distances from one another. 

Such a case is somewhat different from the one discussed 
in my paper {loc. cit.). Here it is necessary to assume that 
l = d, where / is the length of the doublet and d the distance 
between them, under which conditions the ordinary inverse 
square law must hold true. The attraction between the 
charges in a row is equal to 



where e is the elementary charge, and the value of k is the 
sum of a series 1 — i + J — i + ^— • • • =0*6931. 

Assuming that the adjacent rows have no effect upon the 
charges, the expression for the surface tension is of the form 



where p — number of particles per unit volume. 
For the normal pressure the expression will be 

* Zeit. Phtjs. Chem. xxxiv. p. 503 (1900). 
t Zeit. Phys. Chem. xxxvii. p. 385 (1901). 



Breaking Stress of Crystals of Rock-Salt. 65 

For rock-salt 

the density = 2*15, 

M= 1*64 x 58-5 xlO- 28 , 

p= ~ =2-24 xlO 22 ., 

Assuming that k is approximately = 0*7, the tension is 
e 1 x 0*7 x 2-24 x 10 22 = 22'1 x 10" 20 X 0-7 x 224 x 10 22 
= 3500 dynes per cm. (approximately). 

The figure obtained is of the same .order of magnitude as 
figures derived by Ostwald and Huletfc (loc. cit.) for barium 
salts. 

The normal pressure P would be accordingly 
P = 98"7 X 10 9 dynes per square cm. 

It is interesting to see now how far the above results 
agree with the experimental evidence. 

An experiment was performed as follows : — 

I took a good specimen of rock-salt crystal and I cut a 
prism of the section about 15 square mm. and about 2-3 cm. 
long. I used a suitable cement to hold the piece from both 
ends, and by applying a suitable weight produced a rupture 
of the crystal into two halves. Measuring the cross- 
section of the rupture accurately, I calculated the weight 
required to produce the rupture per square cm. I have 
repeated the experiment many times with different samples 
of rock-salt. If the crystal is well formed, the agreement 
between individual experiments is fairly good. In one 
series of experiments, I cut the prisms so as to have several 
samples cut parallel to the three principal axes. I have done 
the experiment with three pieces for each direction. For 
one direction I obtained : 

89 lb. per cm. 
103 „ „ 
83 „ „ 

In the other two directions the results were identical. 

In some cases it happened that the rupture took place 
under a much smaller weight. This, however, could be 
attributed either to some faults in the structure of the 
crystal, or to some other disturbances. Such measurements 
were simply disregarded. 

Taking as the average value 91*7 lb., or 41*5 kgrs. per 
square cm., one can calculate the inward pull per row of unit 
length. 

Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. F 



66 Breaking Stress of Crystals of Rock- Salt. 

This is obtained by dividing P by p 1/3 , and becomes 

P 415000x981 1 . ' 
P7* = 2'82xl0? =1*4 dynes per cm. 

If the attraction between tbe charges in a row is not 
appreciably influenced by the adjacent rows, this value will 
represent the accurate value of surface tension in the direction 
coinciding with the vertical axis. If the field is symmetrical 
in all three directions parallel to the main axis, this figure 
will characterize the surface tension of rock-salt in all 
three directions. 

The symmetrical structure indicated by W. H. and W. L. 
Bragg is in accord with the experiment in this sense, but on 
the other hand, the above figure is about 1000 times less than 
one would expect. 

One could expect such a small value if the salt consisted 
of molecules with a very small polarity situated at large 
distances from one another. But in such a case the force 
in the direction coinciding with that of the doublet would 
have to be twice that in the perpendicular direction. This, 
however, is not the case." 

In my paper (loc.cit.) it was shown that in a case of small 
doublets the adjacent rows have practically no influence on 
the attraction between the doublets. 

However, in a case of charges situated at regular distances 
from one another, such seems not to be the case. 

If the charges influence one another, one can expect the 
forces to be weaker in the middle of the substance, and much 
bigger at the surface where the above effect is only one- 
sided. 

It is therefore probable that the above figure 1*4 dynes 
per cm., although quite characteristic for the substance, is 
not the actual value of the force in the surface layer. The 
calculation of these effects is not easy, owing to the fact that 
one has to deal with a very slowly converging series. I 
satisfied myself that these influences may be appreciable, but 
I do not see clearly at the present time whether they can 
account for the weakening of the forces about 1000 times, 
or even more. 

6 Featherstone Buildings, 
High Holborn, 
London, W.C. 1. 



[ 67 ] 

VII. The Measurement of Chance. 
By Norman Campbell, Sc.D.* 

Summary. 

IT is maintained that the chance of an event happening 
is always a physical property of a system, measured 
by a process of derived measurement involving the two 
fundamental magnitudes — number of events and number 
of trials. 

Chances are not measurable by a process of fundamental 
measurement. But the calculation of chances is analogous 
to fundamental measurement. It is usually theoretical, and 
is valuable only in so far as the calculated chances are 
confirmed by measurement. 

When a proposition concerns a system characterized by a 
chance, it may sometimes (but by no means always) be 
regarded as having a definite probability determined by that 
chance. The probability of propositions which do not 
concern systems characterized by chances has nothing: to do 
Avi tli chance. 



1* It is generally recognized that there are two kinds of 
" probability.'-' There is (1) the probability (of the 
happening) of events, and (2) the probability (of the truth) 
of propositions. Etymologically the term belongs more 
properly to the kind second of probability, and it will be 
confined to that kind in this paper. For the first kind, the 
term " chance,''' often used in some connexions as a synonym 
of probability, is available. Accordingly we shall speak 
throughout of the chance of an event happening and of the 
probability of a proposition being true. 

Various opinions have been entertained concerning the 
relation between chance and probability and between the 
methods of measuring them. Some have held that chance, 
some that probability, is the more fundamental conception, 
and that the measurement of the less fundamental depends 
on that of the more fundamental conception. Others have 
held that only one of the two, or neither, is measurable. 
The conclusion towards which this paper is directed is that 
chance, in the sense primarily important to physics, is a phy- 
sical property measurable by ordinary physical measurement. 
This view is similar to that held by Venn ; indeed, it is 

* Communicated by the Author. 
F2 



68 Dr. Norman Campbell on the 

probably the view that Venn would have held if he had ever 
considered the nature of: physical measurement. But the 
further view often attributed to Venn, though it is doubtful 
whether he actually held it, that probability is always 
measurable in terms of chance — this view will not be upheld, 
but, so far as it is discussed at all, will be combated. 

Many of the ideas and terms used in the discussion are 
explained more fully in my ' Physics,' to which references are 
made by the letter P. In fact, this paper may be regarded as a 
substitute for pp. 168-183 of that book, some of the difficulties 
of which are avoided by the alternative method of treatment. 
However, I should like to add that I do not accept any of 
the criticisms that have been directed against those pages by 
others. 

2. Suppose we are presented with a pair of dice, and asked 
what is the chance that when one of them is thrown it will 
turn up six. The answer may be different for the two dice. 
If one of them is accurately cubical with its centre of mass 
accural ely at the centre of the cube, while the other has 
corners and edges variously rounded and is loaded so that 
the centre of mass is appreciably nearer one face than 
another, then the answer will be different. On the other 
hand, if in all respects the dice are the same, then the 
answer will be the same ; even if they are both inaccurate 
in form and both loaded — the inaccuracy of form and the 
loading being the same — the chance that they will turn up 
six will be the same. This chance is something uniformly 
associated with and changing with the structure of the die, 
just as is (say) the electrical resistance. This uniform 
association of the resistance with the other characteristics 
of the die is what we assert when we say that; the resis- 
tance is a physical property of the die, and accordingly 
the chance of turning up six is a physical property as much 
as the resistance. 

Moreover, the chance is measured by essentially the same 
process as that by which the resistance is measured. Resist- 
ance is measured (in its original meaning) as a derived 
magnitude by means of a numerical law (P. Ch. xiii.).. We 
place two electrodes in contact with opposite faces of the 
die, and measure the current which flows through it when 
measured potential differences are maintained between the 
electrodes. We then plot current against P.D., and find we 
can draw a straight line through (or more accurately 
among) the resulting points. The fact that the graph is a 



Measurement of Chance. 69 

straight line passing through the origin shows that a 
numerical law of a certain form holds, and therefore that the 
die is characterized by a single definite magnitude, which is 
what we mean by resistance ; the slope of the line tells us 
the numerical value of this magnitude. When we proceed 
to measure the chance of turning up six, we make several 
groups of trials, measure in each group the number of trials 
and the number of those in which six turns up. We plot 
these two fundamental magnitudes against each other, and 
find that a straight line can be drawn through (or among) 
the points. The fact that the graph is straight and passes 
through the origin tells us that the die is characterized 
by a definite magnitude, which is what we mean by chance ; 
the slope of the line tells us the numerical value of this 
magnitude. 

3. The resemblance is exact in all essentials. But as the 
conclusion that chance is an ordinary physical magnitude 
does not seem to be universally accepted, some objections 
may be considered. 

The first may be (though I am not sure that it will be) 
raised by those who denounce the "frequency theory " of 
probability. They might say that, though the derived 
magnitude, estimated in the manner described, is a true or 
approximate measure of the chance, yet it is not what is 
meant by the chance — that is something much more abstruse. 
Such an objection can only be met by stating more clearly 
what is asserted, and recognizing any difference of opinion 
that remains as insoluble. What I assert is (1) that all 
chances determined by experiment are determined by a 
relation between frequencies, and (2) that chances are 
important for physics only in so far as they represent 
relations between frequencies. Few examples can be cited 
in support of (1), for chance in physics is usually a theo- 
retical and not an experimental conception ; but it may be 
suggested that anyone who proposed to attribute to the 
chance of a given deflexion of an a-ray in passing through 
a given film any value other than that determined by fre- 
quency, could convince us of nothing but his ignorance of 
physics. In support of (2) it may be pointed out that 
the chance, which is such an important conception in the 
statistical theories of physics, enters into the laws predicted 
by those theories only because it represents a relative 
frequency. 



70 Dr. Norman Campbell on the 

4. A second objection may be based on the fact that the 
straight line has to be drawn among and not through the 
experimental points. It may be readily admitted that this 
fact shows that the chance cannot be estimated with perfect 
accuracy. But there is also some uncertainty in determining 
the resistance ; and since I am concerned only to enforce the 
analogy between chance and resistance, the admission is 
innocuous. If it is urged that this uncertainty shows that 
the derived magnitude cannot be the chance, because chance 
is something to which a numeral may be attached with 
mathematical accuracy, then it is replied (as in answering 
the first objection) thnt such a chance, to which no experi- 
mental error is attached, is something totally irrelevant to 
physics. 

But the objection may be put in a less crude form. It 
may be urged that, in the matter of experimental error, there 
is a fundamental difference between resistance and chance. 
For in the latter, but not in the former, the error is something 
essential to the magnitude ; we can conceive of a resistance 
measured without error, but not of a chance measured with- 
out error ; if all the points lay accurately on the line, then 
the magnitude measured by its slope would not be a chance. 
Again, there is a simple relation between the average error 
about a point on the " chance " line and the co-ordinates of 
that point ; while in the " resistance " line the relation is 
much more complex, and depends on the exact method of 
measuring the current and potential. All this is quite true, 
and would be important if we were considering the theory of 
chance or of resistance. There is a great difference in those 
theories; we suppose that the "real" Ohm's law holds between 
the real and not the measured magnitudes of the current and 
potential, while there is no real magnitude involved in the 
chance relation. But w T e are not considering theory but 
experiment; I am only asserting that chance is an experi- 
mentally measured magnitude. The fact that the errors in 
the two cases are differently explained does not affect the 
fact that there are errors in both cases, and that the problem 
of determining the derived magnitude in spite of these errors 
is precisely the same. 

5. As a third objection it might be urged that the two 
measurements are not really similar, because ihe chance is 
not really determined by the slope of the line, but by the 
ratio of the two numbers when they are sufficiently great. 
Here is a misconception which it is important to correct. 
If we know that the happening of the events is determined 



Measurement of Chance. 71 

by chance, then it is true that we need only plot one point 
on the line ; and the distribution of the "errors is such that 
the relative error of a determination from a single point is 
less the greater the number of trials involved. We shall 
group all our observations together, so as to make their 
total number as great as possible. But similarly, if we know 
that the material of the die obeys Ohm's law, one observa- 
tion is sufficient to determine its resistance ; and the accuracy 
of the determination will be greatest if we choose the 
measuring current within a certain range. An even closer 
parallel would be obtained if we took in place of resistance 
the derived magnitude, uniform velocity. If we knew that 
the velocity was uniform, we should choose our time and 
distance as great as possible, and determine the velocity from 
this single pair of values without troubling to plot smaller 
values. 

But in order that determination by a single point should 
be legitimate, we must know that the events really are 
determined by chance, and the only test of chance is that, 
when a series of points are plotted in the manner described, 
the only regularity discoverable in them is that they lie 
about the straight line. Their distribution about that line 
must be random. Thus, to take Poincare's excellent example, 
if the trials were made by selecting the first figure of the 
numerals in a table of logarithms in the conventional order, 
and the events were the occurrence of the figure 1, the 
plotted points would lie on the whole about a line with 
a slope of 1/10. But a regularity of distribution about 
that line would be apparent ; we should have a series of 
points all lying above the line followed by a set all lying 
below. If, on the other hand, we took the last figure of 
the numerals, no such regularity would be apparent ; the 
distribution of the points about the line would be random ; 
the events would be dictated by chance. 

It is of the first importance to insist that in measuring a 
chance we are picking out the only regularity that we can 
find in some sequence of phenomena, leaving a residuum 
which is purely random. Randomness is a primary con- 
ception, incapable of further definition ; it cannot be explained 
to anyone who does not possess it. It is based, I believe, on 
observation of the actions of beings acting consciously under 
free volition ; and it is subjective in the sense that what is 
random to one person may not be random to another with 
fuller knowledge (P. p. 203") . There are certain forms 
of distributions that are random to everybody ; it is this 
common randomness, objective in the sense in which all the 



72 Dr. Norman Campbell on the 

subject-matter of science is objective, that is the characteristic 
of the objective c'hance which is physically measurable. 
Chance is applicable only to events which contain an element 
which is wholly and completely random to everybody *. 

6. We shall then base our further discussion on the 
assumption that any physically significant chance (of the 
happening of an event) is a measurable derived magnitude, 
a property of the system concerned in that event, determined 
by a linear numerical law relating the fundamental magni- 
tudes, number of events and number of trials. It is thereby 
implied that the " errors " from the law are random, for 
otherwise the law would not be linear. The definition of 
chance as the limiting ratio of the fundamental magnitudes 
as they tend to infinity is identical with that given, if it is 
known as an experimental fact that the magnitudes of the 
errors fulfil certain conditions which need not be discussed 
in detail here ; these conditions are not inconsistent with 
the randomness of the errors. 

Chance as a fundamental magnitude. 

7. Another important question may be raised, again 
suggested by the analogy with resistance, Resistance means 
the derived magnitude defined by Ohm's law. But actually 
resistance is measured nowadays, not as a derived, but as a 
fundamental magnitude, in virtue of the Kirchhotf laws for 
the combination of resistances in series and parallel f. Can 
chance, though meaning the derived magnitude, be measured 
independently as fundamental ? 

In order that a property may be measured as a funda- 
mental magnitude, it is necessary that satisfactory definitions 
of equality and of addition should be found (P. Ch. x.). 
In addition, some numerical value must be assigned arbi- 
trarily to some one property, which with all others can 

* In this sense the last figure of the logarithm is not wholly dictated 
by chance ; for we know that there must be some regularity in the 
distribution of the points about the straight line, even if we cannot say 
exactly what it is. In the strictest sense, therefore, there is no such 
thing as the chance of the last figure being 1. But there are events 
which are, at present at least, wholly dictated by chance in this sense, 
e. g. the disintegrations of a radioactive atom. Here I do not think 
anyone has imagined what kind of regularity there can be, except the 
falling of the plotted points about the straight line which determines 
the chance. 

f Ultimately measured, that is to say, by the makers who calibrate 
our resistance boxes. In the laboratory we use a method which is 
essentially that of judgment of equality with a graduated instrument. 



Measurement of Chance. 73 

be compared by means of these definitions. Since we take 
the meaning of chance to be that of the derived magnitude, 
the definitions will be satisfactory if they are in accord- 
ance with the derived process of measurement ; but we 
shall not succeed in establishing an independent system of 
fundamental measurement, unless the definitions are such 
that they can be applied without resort to that process. 

The arbitrary assignment is usually made by attributing 
the value 1 to the chance of an event which always happens 
as the result of a trial. The only question that can arise 
here, namely whether all other chances can be connected 
with this chance by addition and equality, will be considered 
presently. The definition of addition presents no difficult}'. 
The chance of A happening is the sum of the chances of 
.r, y, z, ... happening if, x, y, z, ... being mutually exclusive 
alternatives, A is the event which consists in the happening 
of either x or y, or z, .... This proposition is introduced in 
all discussions of chance, but it is often introduced as a 
deduction and not as a definition. The inconsistencies which 
result from such a procedure are discussed in P. pp. 174, 184, 
185. As a definition, it is satisfactory in our sense, for 
measurements by the derived method would show that, in 
such conditions, the chance of A is the sum of the chances of 
x, y, z, ..., and yet it does not presuppose such derived 
measurements. If the points in the derived measurement 
lay on the straight line, this result would be a direct 
consequence of the definition of the derived magnitude ; 
but since they do not, it can be deduced from that 
definition only if some assumption about the distribution 
of the errors is made. The assumption that the errors are 
random would probably suffice if randomness could be 
strictly defined ; since it cannot, the agreement of the 
proposed definition of addition with the results of the derived 
process of measurement must be regarded as an experimental 
fact. The definition is thus precisely analogous to that used 
in the fundamental measurement of resistance, namely that 
resistances are added when the bodies are placed in series. 

The definition of equality is much more difficult; in fact, 
it is the stumbling block of many expositions of the measure- 
ment of chance. For resistances we can say that bodies are 
equal if, when one is substituted for another in any circuit, 
the current and potentials in that circuit are unchanged ; 
that definition does not involve a knowledge of Ohm's law 
and of the derived measurement. The only attempt at an 
analogous definition for chance, of which I am aware, is 
that based on the principle of sufficient reason ; chances 



74 Dr. Norman Campbell on the 

are said to be equal when there is no reason to believe 
that one rather than the other will happen as the result 
of any trial. But what reason could there be for such 
a belief based on experiment ? No a priori principle can 
determine the property of a system, which is an experi- 
mental fact ; we cannot tell whether a die is fair or loaded 
without examining it through our senses. The only experi- 
mental reason I can conceive for believing that one event 
is more likely to happen than another is that it has 
happened more frequently in the past. But if an attempt 
is made to define " more frequently " precisely, the judg- 
ment of equality is inevitably made to depend on the 
derived measurement, and the fundamental process ceases 
to be independent of it. This dependence is often con- 
cealed by the use of question-begging words. Thus, the 
principle of sufficient reason may be reasonably held to 
decide that, in a perfectly shuffled pack of cards, the chance 
that the card next after a heart is another heart is equal to 
the chance that it is a club. But if inquiry is made what is 
meant by a perfectly shuffled pack and how we are to know 
whether a pack is or is not perfectly shuffled, I can seen no 
answer except that it is one in which a club occurs after a 
heart as often as another heart. But, of course, to define 
perfect shuffling in that way is to admit that the criterion of 
equality is based upon the derived measurement of 
" frequency " *. I can find no proposed definition of the 
equality of chances that is both applicable to experimental 
facts and independent of frequency ; and I conclude, therefore, 
that there is not for chance, as there is for resistance, a 
fundamental process of measurement independent of the 
derived. 

8. But there is a further difference to be considered. 
Even if equality of chance could be defined independently, 
there would still be many chances (and those some of the 
most important) which could not be connected with the unit 
by the relations of equality and addition: Any resistance is 
equal to the sum of some set of resistances such that the sum 
of another set of them is equal to the unit or to the sum of 
some set of units. The analogous proposition about chances 



* It is not always realized by those who calculate card chances in 
great detail that in actual play, even among experienced players, the 
shuffling is so imperfect as to distort very seriously the chances of such 
events as the holding of a very long suit. 



Measurement of Chance. 75 

is not true, if the chances are always experimentally deter- 
mined. Consider, for example, the disintegration of a 
radioactive atom within a stated period. There are only 
two alternatives : the atom does disintegrate, or it does not. 
The sum of the chances of these two events is 1, but the 
chances of the two are not in general equal. And neither 
of them can be shown experimentally to be the sum of the 
equal chances of other events such that the sum of some 
other set of those chances is equal to the unit. The definition 
of unit chance together with definitions of equality and 
addition would never permit us to determine such chances ; 
they can only be determined by derived measurement. 

9. Chance is therefore not capable of fundamental 
measurement. Nevertheless the principles of fundamental 
measurement are important in connexion with chance, 
because they are involved in the calculation of chances. 
When we calculate a chance we always assume that it is 
measurable by the fundamental process. Thus if we calculate 
the chance of drawing a heart from a pack, we argue thus : — 
The chance of drawing any one of the 52 cards is equal to 
that of drawing any other. The chance of drawing one of 
the 52 cards is, by the definition of addition, the sum of the 
chances of drawing the individual cards, and, by the 
definition of unit chance, it is 1. Consequently the chance 
of drawing any one card is 1/52. But the chance of drawing 
a heart is the sum of the chances of drawing 13 individual 
cards ; it is therefore the sum of 13 chances each equal to 
13/52, i. e. 1/4. The calculation is perfectly legitimate, so 
long as we know (1) of how many individual events the 
event under consideration (and any other event introduced into 
the argument) is the sum, and (2) that the chances of these 
individual events are equal. (1) does not depend on the 
derived system of measurement, but it does involve a very 
complete knowledge of the event under consideration; (2), if 
it is an experimental proposition at all, must depend upon 
derived measurement. The calculation is often made when 
(2) is not experimental, and when there is no direct know- 
ledge of (1) ; it is then purely theoretical, and the only 
legitimate use that can be made of it is to confirm or reject 
the theory by means of a comparison of the calculated chance 
with that determined experimentally by the derived measure- 
ment. The fact remains that true chance, the property of 
the system, is always and inevitably measured by the derived 
process and not by the fundamental. 



76 Dr. Norman Campbell on the 

Chance and Probability. 

10. It remains to consider very briefly what connexion, if 
any, there is between the chance of events and the proba- 
bilities o£ propositions. 

Probability is usually admitted to be an indefinable 
conception, applicable to propositions concerning which there 
is no complete certainty, and roughly describable as the 
degree of their certainty. It appears to me one of those 
conceptions which are the more elusive the more they are 
studied ; I am quite certain that I do not understand what 
some other writers mean by the term, and am not at all 
certain that I can attach a perfectly definite meaning to it 
myself. The observations that I can offer are therefore 
necessarily tentative. But it is clear, at any rate, that 
probability is not a property of a system and is not physically 
measurable ; any propositions connecting it with chance 
must depend ultimately on fundamental judgments which 
can be offered for acceptance, but cannot be the subject of 
scientific proof. 

There are two kinds of propositions the probability of 
which may plausibly be connected with chance ; and they 
naturally can apply only to systems that are characterized by 
chances. Of the first kind the following is typical : — This 
die will turn up six the next time it is thrown (or on some 
other single and definite occasion). Here (ef. P. pp. 192-200) 
it seems that, if the proposition is really applied to a single 
occasion only, the probability of the proposition must be that 
characteristic of absolute ignorance ; for the assumption that 
anything whatever is known of the result of a single trial is 
inconsistent with the experimental fact that the result of any 
one trial is random. The only exception occurs when the 
event is one of which the chance is so small (or so great) that 
the happening of it (or failure of it) would force us to revise 
our estimate of the chance or to deny that there was a chance 
at all. Of such coincidences, in systems of which the chance 
has been well ascertained, the assertion that they will not 
(or will) occur may be made with the certainty that is 
characteristic of any scientific statement. There is no 
probability. 

On the other hand, it is very difficult to be sure that only 
a single trial is contemplated. For when such statements 
are important, there is always a clear possibility of a consider- 
able number of repetitions of the trial. If this number is 
so great as to permit a dermination, by derived measurement, 
of the chance of the event within some limits relevant to the 



Measurement of Chance. 11 

problem, then it will be found by examination of the use of 
such propositions that their importance depends simply on 
the value of that chauce. If that chance is greater than a 
certain value, the proposition will be true for the purposes 
concerned ; if it is less, it will be false. I cannot myself ever 
find in such propositions any meaning which is not contained 
in the proposition : — The chance that the die will turn up 
six is greater or less than some other chance. Accordingly 
again, there seems no room for a probability which is distinct 
from chance. 

11. Of the second kind of proposition an answer to the 
following question may be taken as typical : — I have two 
dice, of which the chances of turning up six are unequal. 
I throw one, but I do not know which. It turns up six. 
Which oE the two dice have I thrown? 

Here again (P. pp. 185-192), if the question is asked 
of a single throw, it seems to me that the only possible 
answer is simply, I do not know ; except, as before, if the 
throw would be a ei coincidence " with one die and not with 
the other. For, once more, if the events concerned are 
really characterized by chances, it is inconsistent with the 
statement that they are so characterized to assert that, at a 
single trial, the result, if compatible with either of the two 
" causes/' may not happen as the result of either of them. 
If, on the other hand, the throws are repeated (while it is 
certain that the same die is always used), and if they deter- 
mine the chance of one die rather than that of the other, it 
is clearly certain that this die, and not the other, is being- 
used ; a die can be identified by its chance as certainly as 
by its resistance or any other physical property. But 
intermediate between these extremes, there certainly seem 
to be cases in which, though the evidence is not sufficient 
to enable us to assert definitely which die is being thrown, 
we begin to suspect that it is one and not the other. The 
possibility of such a state of mind arises from the fact that 
there is necessarily a finite period during which the 
evidence is accumulating ; it does not arise when, as in 
the usual determination of resistance, the evidence is obtained 
all at the same time. And our suspicion will increase 
generally with the " probability " as estimated by the well- 
known Bayes's formula for the probability of causes. In 
this case it appears to me that there is such a thing as 
probability, determined by but distinguishable from chance, 
and applying to a proposition, and not to an event. But 
I can find no reason to believe that this probability is 



78 The Measurement of Chance. 

numerically measurable in accordance with Bayes' or any 
other formula. 

But in most cases where an attempt is made to apply a 
probability o£ causes, the condition is not fulfilled that it is 
known that the same die is always used. If that condition 
is not fulfilled, the probability, according to orthodox theory, 
depends on certain a priori probabilities which are not chances. 
The problem then ceases to be one of the connexion between 
chance and probability, and thus falls without the strict 
limits of our discussion. 

12. But it is necessary to transgress those limits for one 
purpose.' It has been often urged by philosophers that 
probability is characteristically applicable to scientific 
propositions, which are to be regarded, not as certain, but 
ouly as more or less probable. If this be so, the con- 
ception of chance, being a scientific conception deriving 
its meaning from scientific propositions, must be subsequent 
to the conception of probability, and the order of our 
discussion should have been reversed. Of course I do not 
accept the philosophical view, and perhaps it will be well to 
explain very briefly why I reject it. 

Doubtless there is a sense in which scientific propositions 
are not certain ; but in that sense no proposition is certain, 
so long as its contrary is comprehensible. For if I can 
understand what is meant by a proposition, I can conceive 
myself believing it. I am not perfectly certain either that 
Ohm's law is true, or that (x + a) 2 — a? -\- Zax -\- a 2 : I can 
conceive myself disbelieving either. If I were forced to say 
which I. believe more certainly, I should choose Ohm's law ; 
for I could give a much better account of the evidence on 
which I believe it. A mathematician, of course, would make 
the opposite choice. But it appears to me useless to com- 
pare the " certainties" of two propositions when they are of 
so different a nature that the source of the uncertainty is 
perfectly different. If a proposition is as certain as any 
proposition of that nature can be, and if nothing whatever 
could make it more certain, then it seems to me misleading 
to distinguish its probability from certainty. 

Now, fully established scientific propositions are ecrtainin 
this sense. They are uncertain only in so far as they predict 
If in asserting Ohm's law, I mean (and I think this is my 
chief meaning) that it appears to me a perfectly complete 
and satisfying interpretation of all past experience and that 
other people appear to share my opinion, then Ohm's law is 



Ignition of Gases by Sudden Compression. 79 

perfectly certain, or at least as certain as any mathematical 
or logical proposition. On the other hand, it' it is meant that 
the law will be to me and to others an equally satisfying 
interpretation of all future experience, then I am not 
absolutely certain ; I am only as certain as I can be about 
anything in the future. And it must be noticed that nothing- 
can make me more certain. If I were predicting something 
about a single future occasion, I might in the .course of time 
become more certain; for that future occasion might some day 
become past. But if, as in the case of a scientific law, I am 
predicting something about all future experience, then, since 
the future is indefinite, no amount of additional experience, 
converting finite portions of the future into the past, can 
make me more certain; for there will always remain as much 
future as before. Such uncertainty as there is in the 
proposition is inherent in its nature ; if it were absolutely 
certain, it would not be the same proposition. 



VIII. Experiments on the Ignition of Gases by Sudden 
Compression. By H. T. Tizaed and D. R. Pye *. 

[Plate I.] 

I. TN a previous paper f, it was shown that when a mixture 
JL of a combustible gas or vapour with air was suddenly 
compressed, explosion miglit take place after an interval the 
duration of which depended on the temperature reached by 
the compression. It is known that below a certain tempera- 
ture, called the ignition temperature, no explosion, and no 
very appreciable reaction, takes place under these conditions ; 
and the experiments referred to showed that just above the 
ignition temperature, the delay before explosion occurs may 
be of the order of one second in certain cases, while — in the 
case of hydrocarbons and air — the delay at a temperature 
some 50° above the ignition temperature was very small. 
It was pointed out that the observed ignition temperature 
must not only depend on the properties of the combustible 
substances, but also on the conditions of experiment, and 
particularly on the rate of loss of heat from the gas at the 



* Communicated bv the Authors. 

t H. T. Tizard, "The Causes of Detonation in Internal Combustion 
gines." Proceedings of the N.E. Coast Institution of Engineers and 
Shipbuilder, May 1921. 



80 Messrs. H. T. Tizard and D. R. Pye on the 

ignition temperature. The fact that this has not been fully 
taken into account previously seems to account in some 
measure for the differences in the results obtained by other 
workers. It was further shown that the period of slow 
combustion before explosion took place also depends on the 
properties of the combustible substance, and a theory was 
briefly developed connecting the "delays" observed at 
different temperatures with the effect of a rise in temperature 
on the rate of combustion, i. e. with the so-called tempera- 
ture coefficient of the reaction. The object of the experi- 
ments described in this paper was to test these theories 
quantitatively, and to attempt to deduce from the results the 
temperature coefficients in certain typical cases. The mea- 
surement of the temperature coefficients of simple gaseous 
reactions is of considerable importance in connexion with 
the theory of chemical reactions,, for one of the great 
difficulties in the development of theory hitherto has been 
the fact that most gaseous' reactions have to be investigated 
under conditions which are complicated by the disturbing 
influence of solid catalysts or of the walls of the containing 
vessel. Gaseous reactions which occur on sudden com- 
pression are free from this complication, for the walls of the 
containing vessel are much lower in temperature than the 
gas; by quantitative measurements of the rate of loss of 
heat near the ignition temperature, and of the delay before 
explosion occurs, it therefore seems possible to gain some 
real insight into the mechanism of homogeneous gas reactions. 
Experiments of this nature also have a considerable practical 
interest for the development of internal combustion engines, 
for, according to our views, the tendency of a fuel to 
detonate at high temperatures depends not only on its 
ignition temperature but also on the temperature coefficient 
of its reaction with oxygen. 

Previous experiments on the ignition of gases by sudden 
compression have been made by Talk, at Nernst's suggestion 
(J. Amer. Chem. Soc. xviii. p. 1517 (1906), xxix. p. 1536 
(1907)), and by Dixon and his co-workers (see Dixon, Brad- 
shaw & Campbell, Journ. Chem. Soc. 1914, p. 2027 ; and 
Dixon & Crofts, p. 2036). Nernst first put forward the view 
that at the ignition temperature the evolution of heat due to 
the reaction was just greater than that lost to the sur- 
roundings; but this suggestion has not hitherto been carried 
further, since no previous workers have attempted to 
measure the rate of loss of heat near the ignition tempera- 
ture. Further, previous work has been mainly confined to 
the measurement of ignition temperatures of mixtures of 



Ignition of Gases by Sudden Compression. 



81 



hydrogen, oxygen, and an indifferent gas. In such experi- 
ments the interval which occurs at the lowest ignition 
temperature between the end of the compression and the 
occurrence of ignition is very small ; under these conditions 
an apparatus of the kind used by Nernst and Dixon gives 
fairly satisfactory results. It is not well suited, however, 
for experiments with other gases, such as the hydrocarbons, 
when there may be an appreciable delay before ignition 
occurs. In such cases it is of great importance to ensure 
that the cylinder in which the compression is effected is as 
gas-tight as possible, and that the piston is held rigidly in 
position at the end of the compression stroke. 

II. The apparatus used for our experiments was originally 
designed and built by Messrs. Ricardo & Co. with a view 
to determining the temperatures of spontaneous ignition of 
various fuels used in internal combustion engines under 
conditions which correspond closely with those obtaining in 
an engine cylinder. 



ria 1 




Fio-. 1 shows diagrammatically the arrangement of the 
mechanism. A very heavy flywheel A rotates quite freely 
on the shaft B, and is kept spinning by an electric motor at 
about 360 R.P.M. The shaft B carries between bearings 
the crank D, and outside one bearing, the internal expanding 
clutch C, which can engage with the flywheel rim. 

The piston E moves vertically in the jacketed cylinder F, 
which has an internal diameter of 4J inches and can be 
raised or lowered bodily in the heavy cast-iron casing of the 
apparatus when the compression ratio is to be altered. The 
length of stroke of the piston is 8 inches, and its motion is 
controlled by the two hinged rods Gr and H of which the 
latter is carried on a fixed bearing at K. The hingo L is 

Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. G 



82 Messrs. H. T. Tizard and D. R. Pye on the 

linked up with the crank pin by the compound connecting 
rod N. That part of the connecting rod attached to the 
crank pin is tubular and contains the sliding rod M attached 
to the hinge L. A clip carried on the sleeve can engage 
with a collar on the inner rod and hold the latter rigid in 
the tube. With the connecting rod locked as one link, the 
crank is rotated by hand for setting the piston in its lowest 
position. When a compression has to be made, the clutch is 
suddenly expanded by a hand lever while the flywheel is 
running at high speed, clutch and crank are carried round 
with the flywheel, and the toggle joint ELK is straightened 
until the hinge L lies on the vertical line between the piston 
centre and hinge K. At the moment when L is vertically 
over K it comes up against a leather pad, and a clip comes 
into action which holds it in this position. At the same 
moment, too, the clip releases the two parts of the com- 
pound connecting rod, so that while the two rods G and II 
are held in the vertical position to take the large downward 
thrust of the piston when explosion of the compressed 
mixture occurs, the crank, clutch, and flywheel are free to 
go on rotating, and the shock due to destruction of the 
momentum of the moving parts is reduced to a minimum. 
The initial temperature of the gases in the cylinder can be 
varied by means of a water jacket round the cylinder, and 
the variation of pressure during and after compression is 
recorded by means of an optical pressure indicator of the 
Hopkinson type. 

For the purposes of the present experiments it was 
necessary : — 

(1) To make the compression space above the piston 

absolutely air-tight so as to eliminate all pressure 
drop due to leakage. 

(2) To arrange an accurate timing gear upon the 

indicator so that pressure-time records correct to 
about one per cent, could be obtained. 

(3) To insert some kind of fan in the compression space 

so that the effect of varying turbulence in the 
compressed gas could be determined. 

(1) Air-tightness of the Compression Space. 

There were three points at which appreciable leakage was 
liable to take place : — 

(a) Round the sides of the main piston. 

(b) Past the indicator piston. 

(c) Round the fan spindle. 



Ignition of Gases by Sudden Compression. 



83 



Of these, the first two were completely eliminated and the 
third reduced to something quite negligible. Figures of 
the actual leakage are given below. 

The method of eliminating piston leak will be understood 
from fig. 2. The piston is made in two parts. Below the 




F/a 2 



cast-iron top, which carries a single piston ring of the usual 
type, there is an aluminium body which for some distance is 
of smaller diameter than the cylinder. Round this waist 
are two cup-leathers, C and D, each with the periphery 
turned upwards, which are separated by a cast-iron ring of 
square section (E, fig. 2). The whole space above and 
between the cup-leathers was filled with castor oil so that 
no air could leak down past the piston head until it had first 
made a space for itself by forcing castor oil down past the 
two cup-leathers in series. 



¥3r 



g 






.JP2 



i A 



na 3 



To eliminate le k past the indicator piston, the plan was 
hit upon of turning the whole indicator upside down and 
then pouring in a little castor oil into the space above the 
piston, as illustrated in fig. 3. Here A represents the piston 
of the inverted indicator with the sp;ice above it filled with 

G2 



84 



Messrs. H. T. Tizard and D. R. Pye on the 



castor oil. The duct BC in the bracket which carries the 
indicator is based on the slope to prevent the possibility of 
any oil getting back along it and into the combustion space 
at C, This arrangement was very satisfactory, for, besides 
forming a perfect seal, it served to keep the piston well 
lubricated ; it was found, moreover, that only a very small 
quantity of oil was forced down past the piston, even after 
continued exposure to high pressure. 

The arrangement of fan and fan-spindle is shown in 
fig. 4. Here the plug A, shown in section, was made to 
screw vertically* downwards in the centre of the cylinder 



LEA THEH 




head, its lower end finishing flush with the top of the com- 
pression space. At the highest compression ratios this space 
is \ inch deep, so that the piston head under these conditions 
is close up to the under side of the fan. The fan was driven 
at varying speed by an electric motor through the usual 
form of flexible drive used for speedometers. 

To prevent leakage up round the fan-spindle, this was 
made long and thin, and was also provided with a cup-leather 
of which the tightness could be adjusted by the screw B in 
fig. 4. By using a motor with ample reserve of power for 
driving the fan, it was possible to keep the cup-leather so 
tight as to reduce the leakage to a negligible amount. 



Ignition of Gases by Sudden Compression. 85 

Experiments on Air Leakage from the Cylinder. 

Observations were made of the rate of fall of pressure 
beginning 10 seconds after compression had occurred. By 
this time all pressure-fall due to heat-loss has ceased, and the 
observed fall represents leakage only. 

Experiments were made — 

(a) With fan stationary. Compression ratio 9 : 1. 

Contents of cylinder were the products of com- 
bustion of a benzene and air mixture. 

rp. Deflexion on Indicator 
e * Screen. 

10 sees, after compression and combustion 063 inches. 

20 „ „ „ 0-61 „ 

30 „ „ „ 0-57 „ 

45 „ „ „ 0-56 „ 

60 „ „ „ 0-53 „ 

The indicator calibration was 1 inch deflexion = 188 lb. 
per sq. in., so that the above gives a rate of fall of 

0-1 X 188 >aftll 

— = -5b lb. per sq. in. per sec. 

(b) With fan rotating at full speed, 2000 r.p.m. Com- 

pression ratio 10 : 1. Contents of cylinder pure 
hydrogen. 

m . Deflexion on Indicator 

Time. c 

Screen. 

10 sees, after compression 066 inches. 

20 „ „ 053 „ 

30 „ „ 0-40 „ 

This, owing to the low viscosity of hydrogen and the high 
compression ratio, was a very severe test. The rate of fall was 

'26 x 188 AK „ . xt ■ .u- 

— r = 2*45 lb. per sq. in. per sec. JNow in this expe- 

riment the rate of fall of pressure due to cooling was as high 
as 600 lb. per sq. inch per second immediately after com- 
pression, so that the effect of. gas leakage on the apparent 
rate of cooling was clearly negligible. 

The Indicator. 

It will be convenient at this point, before going on to 
describe the timing apparatus, to give some further data as 
to the indicator used. As stated above, this was of the 



86 Messrs. H. T. lizard and D. R. Pye on the 

standard Hopkinson optical type *, and need not be described 
in detail. The piston used throughout the experiments was 
one of 0*125 sq. in. area. Pressure on the piston deflects 
a spring attached to it, and thus tilts a small mirror which 
reflects a point of light from a fixed lamp. The magnitude 
of the pressure is thus arranged to correspond to the vertical 
(downward) deflexion of the image of the light on a 4£ x 3^ 
photographic plate carried in a camera fixed to the indicator. 
The calibration of the pressure scale was made in two 
ways. Firstly, by subjecting the piston to known oil 
pressures and measuring deflexions on the camera screen, 
and, secondly, as a check on the piston area, by direct dead- 
weight loading of the spring and measurement of the 
deflexion. The oil pressures in the first calibration were 
produced by a carefully gauged vertical plunger loaded with 
weights. The calibration by this method was carried out at 
the beginning of the experiments, and it was found that the 
relation between pressure and deflexion was given very 
closely by a straight line at a slope corresponding to a 
pressure rise of 188 lb. per sq. in. per inch deflexion. At 
the conclusion of the experiments this calibration was 
checked by the dead-weight calibration as follows :; — 

Keading on Camera Screen. 

Load on Spring. Loading. Unloading. 

0-39 in. 040 in. 

Weight carrier = 325 lb 0-54 „ 0-54,, 

Carrier + 10 = 13-25 „ 0*975 „ 0"99 „ 

„ + 20 =23-25 ,', 1-40 „ 143 „ 

„ + 30 =33-25 „ 1-81 „ 1-81 „ 

This also shows a satisfactorily straight-line relationship 
which checks remarkably well with the previous calibration, 
for '^^'9 

Dead weight per inch deflexion = , ~ — 234 lb. 

corresponding to a pressure on a piston of area 0*125 sq. in." 
of 8 x 23-4 = 187-2 lb. 

The previous figure of 188 lb. per sq. in. per inch de- 
flexion has been used throughout the calculations. 

(2) The Timing Gear. 

The pressure in the cylinder being given by the vertical 
motion of a point of light on a photographic plate, it was 
necessary for the accurate measurement of the lengths ot 
the delay period to give the point of light at the same time 

* Hopkinson, Proc. Inst. Mech. Eng. Oct. 1907, p. 863. 



Ignition of Gases by Sudden Compression. 87 

a uniform and exactly known motion across the plate 
horizontally. To obtain this uniform horizontal velocity, the 
arrangement in fig. 5 was adopted. 

A is a vertical cylinder closed at the bottom and carrying 
at the top a guide for the piston rod B. At the lower end 
of the rod B is a light loosely fitting piston in which are 
drilled one or more small holes. The rod B has a collar C 
above the guide, and can be loaded with weights, up to 
50 lb. if need be. The cylinder is filled with paraffin 
which, when the piston and weights have been raised and 
then released by a trigger arrangement, is forced at high 
velocity through the holes in the piston from the under to 
the upper side. This arrangement gives a velocity of fall of 




the piston which is uniform to within 1 or 2 per cent., and 
it was found, moreover, that this velocity varied directly as 
the square root of the weight carried : a result which shows 
that the viscosity of the oil is a negligible factor in deter- 
mining the rate, and that the latter will therefore not be 
affected by any small changes of temperature which might 
occur from day to day in the oil. 

This point was checked experimentally, and it was found 
that the maximum change in the rate of fall produced by 
heating the oil from 19° C. to 65° 0. was 1|- per cent. As 
the oil temperature during the experiments never varied by 
more than 2° or 3°, the effects of temperature changes were 
quite negligible. 

Piston and weights are suspended by a steel wire from an 
arrangement of pulleys (F) carried on spindles attached to 






88 Messrs. H. T. Tizard and D. R. Pye on the 

the cylinder head. D and E are balance weights to keep 
the wires taut, and F is a compound pulley which reduces 
the horizontal motion o£ the wire Gr to about one-half the 
vertical movement of the piston. At H the wire Gr is 
divided, and each half wraps round the periphery of a sector 
attached to the arm K of the indicator. The barrel of the 
indicator is thus uniformly rotated through a sector about a 
vertical axis, and the speed of the point of light horizontally 
across the photographic plate in the indicator camera can 
be varied between wide limits by alteration of the weights 
and the number of holes in the loaded piston through which 
the paraffin is forced. Actually it was found in these 
experiments that only two speeds were required, of 5*77 and 
4* 30 cm. per second. These were obtained when the piston 
carried weights of 40 and 24 lb. respectively, and was 
pierced by a single hole £ inch diameter. 

Both the speed and its uniformity were measured by in- 
terrupting the light at known intervals while it traced a 
straight horizontal line across, the plate. To obtain the 
uniform interruptions, 60 equally spaced contacts were 
arranged round the periphery of a disk about 6 inches in 
diameter. An electric motor was used to rotate the disk 
through a friction clutch, the speed being kept steady by 
the operation of a governor, which caused the disengagement 
of the clutch when the speed tended to increase. The 
arrangement was very similar to that of a gramophone 
motor, except that the speed of the latter is kept steady by 
means of a little brake which is operated by a governor, 
whereas in our experiments the governor operated the clutch. 
This apparatus, which was made at the Royal Aircraft 
Establishment, Farnborough, was lent to us for these experi- 
ments by kind permission of the Director of Research, Air 
Ministry. 

To calibrate the falling weight apparatus, the speed of 
revolution of the disk was first adjusted roughly to one com- 
plete revolution in 2 seconds. It was then timed repeatedly 
over twenty complete revolutions by stop-watch ; the mean 
time for this number of revolutions was found to be 36*5 
seconds, the variations in successive timings not exceeding 
•4 second. Hence, since sixty contacts were made each 
revolution, the time between the beginnings of successive 

36*5 

contacts was -=7* ztt; = 0*0304 second. The 6-volt lamp of 

60 x 20 r 

the indicator was then connected up, through the contacts 

on the disk, to a 12-volt battery ; it was thus greatly 

" overrun " whenever contact was made. The effect of 



Ignition of Gases by Sudden Compression. 89 

overrunning the -lamp in this way was to make it flicker 
brightly, so that when the indicator mirror was rotated by 
the dropping weight, the reflected light made a horizontal 
line of dashes across the photographic plate, the interval 
between the middle of two consecutive dashes corresponding 
to the interval between two contacts on the uniformly 
rotating disk. The magnitude and uniformity of the velocity 
of rotation of the indicator mirror by the falling weight 
could thus be measured. Records were taken with falling- 
weights of 8, 24, and 40 lb. 

To show the uniformity of motion during descent, the 
results of measuring two plates taken with the 24-lb. load 
are given in full. The apparent variations in any single 
experiment are due rather to the difficulty of estimating the 
centre of a flicker to 0*1 mm. than to any real variation in 
the rate of fall of the loaded piston. 

Plate B. 

0-65 cm. 
1-30 
1-95 
2-6 

3-28 
395 
4-58 
5-22 
585 
651 
7-12 
7-78 
8--10 
9-06 
9-70 

Mean 1 flicker in 0132 cm. Mean 1 in 0*129 cm/ 

1 flicker = •0304 second. 
. * . 1 second = 4*34 cm. on plate. 1 second — 4*26 cm. 

Mean 4'30 cm. = 1 second. 

Similar records with loads of 8 and 40 lb. gave time 
scales of 

2*22 cm. = 1 second 
and 5*77 cm. = 1 second. 

The means for the two plates with 40-lb. load were 5*75 and 
5*79 cm. per second, so that it may safely be assumed that 
the time scale was known correctly to within 2 per cent., and 
probably less. This is a degree of accuracy as great as 
that with which,, as a rule, it is possible to measure the 
records. 

The vertical movement of the weights was about 7 inches, 





Plate A. 


5 flickers in 0*70 


10 


1-34 


15 


2-0 


20 


266 


25 


330 


30 


3'98 


35 


463 


40 


529 


45 


5-93 


50 


659 


55 


7-22 


60 


7-90 


65 


8-53 


70 


9-21 


75 


9-89 



90 



Messrs. H. T. Tizard and D, R. Pye on the 



which produced a horizontal movement of the point of light 
more than twice as great as was necessary to traverse the 
photographic plate, so that ample time was provided for the 
speed to have become uniform before a pressure record was 
taken. The fact that the indicator and all gear except the 
actual falling weight was carried on the cylinder head, made 
it possible to raise or lower this when changing the com- 
pression ratio without affecting the timing gear. The only 
difference was an alteration of about an inch in the distance 
fallen by the weights between the position for maximum and 
minimum compressions. 

Figs. 6 and 7 are prints taken from two typical records. 

Fiff. 6. 



I 

J 






OS 

7iW 



Secon3s 



o-ys- 



In the first there is no explosion, and the record shows simply 
the rise of pressure due to compression, and subsequent fall 
as the compressed mixture cools. Fig. 7 shows the first 
rise of pressure due to compression, the " delay " period of 
nearly constant pressure followed by the practically instan- 
taneous rise on explosion, and finally the rapid cooling of 
the intensely hot products of combustion. It may be 
mentioned that the spring of the indicator was protected from 
the force of the explosion pressure by a stop which prevented 
the deflexion from ever being greater than just to the edge 
of the photographic plate. 

J II. If a gas at a known temperature and pressure is 
suddenly compressed in a gas-tight cylinder, we can calculate 
from a measurement of the maximum pressure reached the 
average temperature of the gas at the moment of maximum 



Ignition of Gases hy Sudden Compression. 91 

compression. If the volume then remains constant, measure- 
ment of the rate of fall of pressure with time gives the rate 
of loss of heat, if the specific heat of the gas is known. One 
would expect this rate to be closely proportional at any 
moment to the difference in the average temperature of the 



Ficr. 7. 



£ 400 



£ 300 



.C; 200 





(B.I.) /ffnit/on 
Max/mum 

Oe/ay=t 
fa/7 



compre ?s/o/? 
J? ■ 



/W 



Te/r?p.3/0°C. 
seconds, 
speeo'. 



0-5 
Time in Secor/o's. 



gas and that of the cylinder walls. If the temperature of the 
gas is T° absolute, and that of the walls is 0, then 



§=<*-">> 



(i) 



where " a " is a constant which we call the cooling factor, 
and which depends on the nature of the gas, and its degree 
of turbulence in the cylinder. The results given later will 
be seen to justify this equation. 
Integration of (1) gives 



i T o -0 



(2) 



where T is the average temperature when t seconds has 
elapsed from the moment the average temperature was T . 



92 



Messrs. H. T. Tizard and D. P. Pye on the 



Since the total volume remains constant, this equation can be 
written 



kg." 



f 



P-P, 



at 



(?) 



f 

if the simple gas laws hold. P max . is here the maximum 
compression pressure, P the observed pressure t seconds after 
the maximum compression pressure is reached, and P/- the 
final pressure of the gas when its temperature is the same as 
that of the walls. P/is therefore equal to rxPi, where r is 
the compression ratio and P x the initial pressure (in these 
experiments one atmosphere) before compression. 

Fig. 6 is a typical cooling curve obtained when air at an 
initial pressure of 14*73 lb. sq. in. and temperature of 
23° C. was compressed in the ratio of 7*02 to 1. The time of 
compression was 0*08 second, and the values of the cooling 
factor a, obtained from observations of the fall in pressure, 
are given in the following table : — 

Table I. 



Time after 


Observed 




P -P* 

, -""max. *-f „ 

lo s p p* ~ s - 


2-3S 


max. pressure 


pressure = P 


P-fy 


a =~T' 


(seconds). 


lb./sq. in. 




r r f 







199-7 


96-3 






•173 


185-7 


82-3 


•068 


•90 


•347 


174-6 


71-2 


•131 


•87 


•52 


165 


61 6 


•194 


•86 


•695 


158-3 


54-9 


•244 


•81 


•87 


152-4 


49 


•294 


•78 


Me? 


m=-85 




6 (initial temp.) = 




Pi = 14-73 lb./ 


23° C. 


Pf=PiX 702 =103-4. 


sq. in. 


Max. temp, (calc.) = 




299° C. 





It was found as a rule that the calculated cooling factor 
tends to diminish as " t' 3 increases. This may be due in part 
to errors of observation, for the errors are big when the time 
interval after the attainment of maximum pressure is small : 
but experimental error will not wholly account for it, and it 
may be explained on the reasonable assumption that there is 
a fair degree of turbulence in the gas just after a sudden 
compression, which dies down after a short time. Potation 
of the fan at a high speed increases the rate of cooling con- 
siderably ; experiments with air gave a cooling factor about 



Ignition of Gases by Sudden Compression. 93 

three times as great when the fan was running as when the 
gas was " stagnant." This will be referred to later when 
discussing the results of the experiments of ignition. Raising 
the compression ratio also increases the cooling factor for 
air. This is also to be expected since the distance between 
the top of the piston and the head of the cylinder is lessened. 
The cooling factor for " stagnant " air in the apparatus was 
found to increase from about 0*76 at a compression ratio of 
6 to 1 to about 1*0 at a compression ratio of 10 to 1, the 
distance between the piston and cylinder head being approxi- 
mately 4 cm. (1*6 in.) at the lower and 2*3 cm. ('9 in.) at the 
higher compression ratio. The experimental error in the 
cooling factors, obtained by experiments similar to that 
quoted above, is probably about 5 per cent. It should be noted 
that the rate of loss of heat under the above conditions is 
considerable. For instance, in the experiment quoted the 
maximum difference in temperature between the gas and the 
walls after compression is 276° ; and since the specific heat 
c v is about 0*18, the air is losing heat at the maximum 
temperature at the rate of 

0'18 x '85 x 276 = 42 calories per second per gram. 

It follows that if an explosive mixture of gases is suddenly 
compressed to its ignition temperature, in such an apparatus 
as that described above, the initial rate of the chemical 
reaction at the lowest temperature at which ignition is ob- 
served must be considerable, for the evolution of heat due to 
the reaction must equal approximately the rate at which heat 
is lost to the walls. For example, the total heat of com- 
bustion of a mixture of a paraffin hydrocarbon and air, in the 
correct proportions to burn to C0 2 and water, is approximately 
700 calories per gram of mixture. If it ignites when suddenly 
compressed to such a temperature that the rate of loss of heat 
is 35 calories per gram, the reaction, if it continued uniformly 
at the initial rate, would be complete in 20 seconds. This 
illustration may serve to show the general nature of the 
reactions that occur on sudden compression ; what occurs in 
practice is that the gas, or part of the gas, reacts so that the 
evolution of heat takes place at a somewhat higher rate than 
the loss of heat by conduction etc. ; hence the temperature 
of the reacting gases must automatically increase, and with 
it the rate of reaction, until the gas " explodes." The interval 
between the end of compression and the explosion must clearly 
depend mainly on three factors : (a) the compression tempera- 
ture, (b) the temperature coefficient of the reaction, (c) the 
rate of loss of heat to the walls. 



94 



Messrs. H. T. Tizard and D. B. Pye on the 



IV. It is now generally recognized that the rate of an 
ordinary chemical reaction varies with the temperature in a 
way which may be empirically expressed by an equation of 
the form 

_B 

k=Ae T , (4) 

where k is the velocity constant, and A and B are constants, 
B being the temperature coefficient. 

In the case of reactions evolving heat, we can write this : 

_B 

Q=A« T , (5) 

where Q is the initial rate of evolution of heat when a definite 



mixture is suddenly compressed to the temperature T. 
the lowest ignition temperature T we have, 

B 



At 



Qo = A? 



(6) 

where Q can be measured from observation of the rate of 
cooling of the gases at a temperature slightly below the 
ignition point : i. e., 

Qo=«*.(T»-0), (7) 

where a is the cooling factor, c v the specific heat of the 
mixture, and 6 the temperature of the walls. 

Now suppose that the gases are compressed initially to a 
higher temperature T : the initial rate of loss of heat will 
then be higher than Q , namely 

but since the effect of temperature on the rate of evolution 
of heat due to the reaction is so much greater than that on 
the rate of loss of heat, it is sufficiently accurate at present to 
assume for our purposes a constant initial rate of loss of heat 
= Q . The initial rate of reaction at the higher temperature 
will be given by 

Q T = A<?~ 7E \ 
From (5) and (6) we have 

^'ii 1 -^- • • • - (8) 

where b = 0*4343 B. 

Now under these circumstances the net rate of gain of heat 

is (Qt — Qo) calories per second. 



Ignition of Gases by Sudden Compression. 95 

The initial rise of temperature is therefore given by 
dT „ ^ r, /Qg 



§-Q.-Qo=Qo(J-i) ... (9) 



/n 1 dT=^dt, 

(Ht _ -, \ c v 



or 






But since Q = ue v (T — 0), 

we have 



7Q^-<|) = a (VV- 



(10) 



V. Can this equation be used to determine the delay, or 
period of slow combustion, that occurs before the temperature 
suddenly rises very rapidly, i. e. before explosion takes 
place ? Strictly speaking, this could only be done if Q, the 
rate of evolution of heat during the initial reaction, depended 
only on temperature and not on the concentration of the 
reacting substances. According to all ordinary theories of 
reaction, this would not be true ; the rate of reaction should 
depend in some way on the concentration of the reactants. 
If we consider such a reaction as the combustion of heptane 

C 7 H 16 + 110 2 = 7C0 2 + 8H 2 0, 

we can hardly suppose it is necessary, before the initial re- 
action — whatever is its nature — can occur, for 1 molecule of 
heptane and 11 molecules^ of oxygen to collide ; but it is 
reasonable to assume that the rate of reaction must at least 
be proportional either to the concentration of the heptane or 
to that of the oxygen, or to the product of the two. It is 
necessaiw, however, to point out that there is little, if any, 
satisfactory evidence that homogeneous reactions in gases 
obey what is ordinarily understood by the law of mass action. 
In fact, evidence from the ignition of gaseous mixtures by 
sudden compression points rather to the reverse. Dixon 
and Crofts' experiments * on the ignition of mixtures of 
hydrogen and oxygen are difficult to explain by any reasonable 
assumption as to the mechanism of the reaction based on the 

* Dixon & Crofts, Trans. Chem. Soc. 1914, p. 2036. 



96 Messrs. H. T. Tizard and D. R. Pye on the 

law of mass action, even after taking into account the possible 
effects of a different rate of loss of heat in the mixtures with 
which they experimented ; the authors state, in fact, that they 
"can offer no satisfactory explanation of the phenomena 
observed." Recent experiments by one of us * have shown 
that the ignition temperatures of hydrocarbon-air mixtures 
are independent, within the errors of measurement, of the 
proportion of the hydrocarbon within quite wide limits. For 
instance, when the proportion of heptane to air was varied in 
the ratio of 1 to 10, the ignition temperature was only lowered 
apparently by some 8° C. from 293° C. to 285° C. As the 
error of observation may certainly amount to 4° C. in such 
experiments, there is here no evidence that the alteration of 
the concentration of heptane has any effect on the ignition 
temperature. If, as might appear reasonable on the law of 
mass action, the rate of reaction depended directly on the 
concentration of the heptane, we should expect it to be 10 
times as great in one case as in the other. Now, it is shown 
later in this paper that the temperature coefficient B, in the 
case of heptane, is about 13,000, from which it follows from 
equation (5) or (8) that if the initial rate of reaction on com- 
pression were really 10 times as great in one case as in the 
other, the ignition temperature of the richer mixture should 
be over 50° lower than that of the weaker mixture. One 
could not fail to detect with certainty a difference of this 
order : hence from the experimental results we come to the 
conclusion that the rate of combustion under these conditions 
does not depend on the concentration of the heptane vapour 
within wide limits. Experiments with other similar sub- 
stances support this view. 

YI. If this be correct, the only probable alternative is that 
the rate depends essentially on the concentration of the 
oxygen. We have not yet attempted to put this to a direct 
test in the apparatus used by us, since it would be necessary, 
to do this completely, to explode detonating mixtures of 
hydrocarbons and oxygen ; and we were anxious to avoid 
the danger of breaking the apparatus before other important 
experiments were carried out. We intend, however, to 
examine this point in the near future. 

It remains to consider whether, if the rate of reaction is 
directly proportional to the concentration of oxygen, the 
effect of the automatic decrease in oxygen content on ignition 
can be safely left out of account in calculating " delays " by 

* Tizard (he. oit.). 



Ignition of Gases by Sudden Compression. 97 

means of equation (10). For this purpose we have to estimate 
to what extent the reaction occurs during the slow period of 
combustion before the explosion occurs, and the pressure- 
time curve becomes almost perpendicular to the time axis 
(see fig. 7). Taking the experiments on heptane as an 
example, we find (Table III.) that the biggest delay observed 
was about 0'6 second ; that the ignition temperature under 
these conditions is 250°C. = 553 absolute, and that the differ- 
ence of temperature between the gas and the walls was about 
210° C. The observed cooling factor was 0*51. Since the 
specific heat of the mixture experimented with is approxi- 
mately = 0*2, the rate of reaction at the ignition temperature 
must correspond to a heat evolution of 

0'51 X 0"2 x 240 = 24 calories per gram per second. 

The total heat of combustion of 1 gram of the mixture 
(containing about 75 per cent, of the theoretical amount of 
heptane for complete combustion) is about 510 calories, so 
that it is evident that during the period when the temperature 
is only rising slowly, which is always less than half a second, 
the amount of reaction and therefore the changes in con- 
centration must be small. Once the temperature begins to 
rise quickly, it is evident that the disappearance of oxygen 
can have only a secondary effect on the rate of the reaction 
compared with that of the rise of temperature until the com- 
bustion is nearly complete, so that the error involved in the 
calculation from equation (10) of the total time of combus- 
tion, by ignoring the effect of changing concentration, must 
be small. In view of the unavoidable experimental errors in 
carrying out experiments of this kind and of our still in- 
complete knowledge of the mechanism of combustion, we do 
not, in fact, think that any attempt to take fully into account 
such secondary effects is justified at present. 

VII. We therefore arrive at the conclusion that the time 
for complete combustion of an explosive mixture of gases 
when suddenly compressed to a temperature above its ignition 
temperature is closely given by the integration of equation 
(10), and is therefore 

t = — fh- - I at-^ d 



-(?} ' ' (11) 



1 p 00 1 

J-o /Jm ( > 



where T is the lowest ignition temperature under the con- 
dition of the experiment, and T is the initial compression 
temperature. 

Phil. May. Ser. 6. Vol. 44. No. 259. July 1922. H 



98 Messrs. H. T. Tizard and D. R. Pye on the 

In this equation, -^ is given by equation (8); its value 
Wo 
depends on the magnitude of the temperature coefficient of 
the reaction. It is not possible to integrate the equation 
completely, but the integration can be carried out by approxi- 
mate methods for any value of ^- . In the table below are 



To 



given values of the expression a (-^n — )t a t different 

T l ° ' b 

values of ^ , and for various values of m . The 

i. ° i ° 

corresponding curves are shown in hg. 8. 

Fig. 8. 




If our views are correct, these should form standard curves 
representing the delays which should be observed under 
different conditions when any explosive mixture of gases is 
compressed to a temperature above its lowest ionition 
temperature. The application of the theory to any specific 
case should enable the temperature coefficient of the reaction 
to be determined. 




Ignition of Gases by Sudden Compression. 99 

Table II. 
Values of a( ° \t for different values of r . r and . 



T/T . 


6/T =12. 


= 10. 


=8. 


=6. 


= 4. 


1004 


•086 


•112 


•155 


•233 


•418 


101 


•056 


•075 


•108 


•169 


•321 


103 


•0245 


•036 


•056 


•098 


•209 


107 


•0081 


•014 


•025 


•051 


•131 


1-11 


•0033 


•0065 


•0135 


•032 


•095 


1-27 


•0002 


•0007 I 

1 


•0022 


•0083 1 


•0415 



The curves bring out clearly the effect of the two main 
factors which determine the characteristics of an explosion 
by sudden compression ; namely, the initial rate of loss of 
heat, and the temperature coefficient. If two gases have the 
same ignition temperature under the same conditions of loss 
of heat, the sharpest explosion will occur in the case of the 
gas with the. highest temperature coefficient, and the greater 
in this case will be the effect, on the magnitude of the delay 
before explosion, of: a higher temperature of compression. 
On the other hand, in any one case, the ignition temperature 
will be raised by carrying out the experiment under con- 
ditions which involve an increased rate of loss of heat ; at the 
same time the sharpness of the explosion will also be in- 
creased. 

VIII. To test the above theory, and to use it to obtain a 
measure of the temperature coefficient in certain cases, we 
chose three substances : heptane C 7 H 16 , ether C 2 H 5 .O.C 2 H 5 , 
and carbon bisulphide CJS 2 . These substances were chosen 
for the following reasons : (a) they could be obtained in a 
sufficiently pure state ; (b) they all have low ignition tem- 
peratures, which lessens the practical difficulties of the 
experiments ; (<•) they are known to behave very differently 
from the point of view of detonation when used as fuels for 
internal combustion engines; and (d) their difference in 
chemical and physical properties makes the comparison of 
their behaviour on combustion particularly interesting. To 
test the theory adequately we considered it absolutely 
necessary, particularly in view of the simplifying assump- 
tions made, not to be content with one set of conditions for 
the ignition experiments. Two series of experiments were 
therefore made with each substance; in the first series the 

H2 



100 Messrs. H. T. Tizard and D. R. Pye on the 

gaseous mixtures with air were compressed in a non-turbulent 
condition, while in the second a high-speed fan was kept 
running throughout the period of compression and subsequent 
ignition. The use of the fan increased the rate of heat-loss 
at the compression temperature by about three times ; hence 
the difference in ignition temperatures observed with and 
without the fan running gives an important and necessary 
check on the value of the temperature coefficient which is 
calculated from the " delay " curve obtained when the non- 
turbulent gases are compressed. The temperatures given 
below represent the average temperature of the gas at the 
instant of maximum compression. By measuring the com- 
pression pressures in each experiment, a value of ei 7," the 
apparent ratio of the specific heats, is obtained from the 
expression : 

V 

•*- max. V 

where r is the compression ratio. 

The average value of 7 is taken for the series, and the 
compression temperatures then calculated for each case from 
the expression 



T 



y— 1 





In each set of experiments the initial mixture of gases was 
of the same composition throughout, the proportion of air 
being somewhat greater than that required for complete 
combustion. The intention of using a weak mixture was to 
avoid as far as possible the deposition of carbon ; as stated 
above, the absolute ignition temperature is not affected 
appreciably by fairly wide changes in the original strength 
of the mixture. 

IX. The first results with heptane gave a very satisfactory 
confirmation of the theory developed above. The results of 
the experiments are given in the following tables and 
diagrams, which include measurements from all the records 
made under each set of conditions. No unsatisfactory 
records have been discarded. 




Ignition of Gases by Sudden Compression. 101 

Table III. 

Ignition of mixtures of heptane and air. 

Mean apparent value of "7" observed = 1*313. 

Tan stationary. Initial pressure (atmospheric) == 11*8 lb./ 

sq. in. in expts. A x to A 10 , and 14'9 lb./sq. in. in 

expts. A u to A 17 . 





Strength of mixture = 1 grm. 


heptane : 20 


grms. air. 




No. of 


Com pa. 


Initial 


Max. Compn. 


Max. Avge. temp, 
cale. from 


Delay 


Cooling 
factor 


expt. 


ratio. 


temp. 


pressure. 


y=l'313. 


obs. 


= a. 


A x 


5'55 


51° C. 


140-5 lb./sq. in. 


280° C. 


No ignition. 


0-49 


A., 


603 


50-5 


156 


295 


019 sec. 


— 


A 3 


702 


50-5 


192 


323 


0-04 


— 


A, 


8-02 


49-5 


225 


346 


0007 


— 


Ag 


5-55 


46 


141 


273 


No ignition. 


0-53 


A, 


582 


46 


151-5 


281 


0-56 


— 


A 7 


6-23 


46 


163 


293 


0-21 


. — 


A 8 


8-02 


46-5 


227 


340 


Very small. 


— 


A 3 


7 02 


47 


189-5 


316 


0-06 


— 


A 10 


702 


42 


♦166 


307 


0-07 


— 


A u 


606 


58 


174 


324 


005 


— 


A l2 


(V56 


53 


177 


315 


006 


— 


a" 


6-56 


48 


176 


306 


0-12 


— 


A 14 


6-56 


44 


177 


298 


0-18 


— 


A 1B 


6-56 


41 


176 


293 


0-28 


— 




656 


38 


178 


288 


0-25 


— 


A 17 


656 


35 


176 


282 


0-58 





The cooling factors in Table III. were obtained from the 
results of those experiments where no ignition occurred by 
the application of equation (3). The results for A x were as 
follows : — 

Table IV. 
Calculation of cooling factor. 



Time from 
max. pressure. 


Obs. pressure. 
lb./sq. in. 


P " P /- 


^ F -T^- 1 - 


8 





140-5 


58-5 






023 sec. 


134 


52 


■0512 


0-51 


0-47 


127-5 


45-5 


•109 


0-53 


0-7 


123-5 


41-5 


•149 


0-49 


093 


119 


37 


•199 


0-49 


M6 


116 


34 


•236 


0-47 


1-39 


112-5 


30-5 


•283 


0-47 



Mean = 0-49 



P^ = 5*55 



x 14-8 = 82-1 lb./sq. in. 



102 Messrs. H. T. Tizard and D. R. Pye 



the 



The results for plate A 5 were similar, the mean being 
a = 0*53. In th© calculations the figure 0*51 has been 
taken. 



Table V. 
Ignition of mixtures of heptane and air. Fan full speed. 

Initial pressure = 14*86 lb./sq. in. Strength of mixture as 
before. Mean apparent value of 7 = 1*310. 



No. of 


Corupn. 


Initial 


Max. Compn. 


Max. Aver. 


Delay obs. 


Cooling 


expt. 


ratio. 


temp. 


pressure. 


temp. calc. 


factors. 


B x •• 


6-03 


61° C. 


157*5 


310° C. 


16 sec. 




B .. 


552 


59 


139 


291 


No ignition. 


143 


B, .. 


5-73 


59'5 


148 


298 


}J 


1-40 


B 4 .. 


591 


59-5 


154 


304 


55 


1-36 


B 5 .. 


6-13 


59-5 


161 


310 


}J 


T47 


B G .. 


633 


595 


165 


313 


0-13 


— 


b: .. 


6-56 


59 


175 


321 


0-08 


— 


B 8 .. 


701 


60 


188 


336 


005 


— 


B 9 .. 


8-02 


60 


9 


362 


o-oo* 


— 



Mean= T42. 



* Ignited before top of compression. 



X. Considering firstly the results of the experiments 
without the fan running, we find that at the lowest ignition 
temperature of about 280° C, the temperature of the walls 

/HP A\ 

was about 40° C. The expression a ^— ^ (equation 11) 



T, 



has therefore the value — 

0*51 x 240 
553 



0*22. 



Assume that the true ignition temperature under these 

conditions is 280° C. = 553 absolute, and that ^ -10: 

lo 
then by integration of the theoretical equation (see Table II.) 
we obtain the folio wing results : — 



Ignition of Gases by Sudden Compression. 

Table VI. 

Ignition of heptane by air. No fan. 

Lowest ignition temperature T taken as 553 absolute. 



103 



T 



assumed = 10. 



T/T . 


T. 


1° C. 


a( ( \^)t(theov.). 


t (calc). 


1-004 


555 


282 


0-112 


0-51 sec. 


101 


558*5 


285-5 


0075 


0-34 „ 


1-03 


569-5 


2965 


0-036 


01 64 „ 


1-07 


592 


319 


0-014 


0063 „ 


111 


614 


341 


0-0065 


003 ;, 


1-27 


702 


429 


0-0007 


0-003 „ 



The last column contains the theoretical " delays " that 
should occur at compression temperatures given in the third 
column, if the temperature coefficient of the reaction 

corresponds to a value of ^ = 10. 

J-o 









Fig. 9 






















ft 

350 


' I 




Heptane and A 




• 




+ ^ 


?""••— 














"*" N ° ' 9nlt '° n ' 






^ V ^^ < K 


"* ~4- "*" 
















No ignition. 


=> + + 



0-3 

Seconds delay 



The corresponding theoretical curve is shown in fig. 9, 
the experimental points taken from Table III. being marked 
with a cross. The general agreement is all that could be 
desired. 



104 Messrs. H. T. Tizard and D. R. Pye on the 

XL If the value for the temperature coefficient so deduced 
is correct, it should be possible to use it to calculate the 
higher ignition temperature when the fan is used, and also 
the shape of the new delay curve. 

Now, the mean cooling factor with the fan has been shown 
to be 1*4-2. The ignition temperature under these conditions 
is evidently about 310° C. (see Table V.), the temperature 
of the walls being 60° C. Hence the ratio of the rate of 
loss of heat with and without fan at the respective ignition 
temperatures is — 

Qo' _ 1'42 (310-60) 
Qo" 



= 2-90. 



0-51(280-40) 

Hence the new (theoretical) ignition temperature T ' (with 
fan) should be given by (see equation (8)) — 

lo g ^ = lo g 2-90=|- o (l-|) = 
= 10 ( 1 -|) 



e. 0-462 



or 



Jr = 0-954. 



T ' = 580 = 307° C. 
This is close to the observed value. 



tv, , J 6 .„ . 5530 

Ine new value ot ™ will be "~^J7T 



9' 5 ; while the value 



of 



(T o '-0\ 1*42x247 



\ T ' ) 



= 0-60. 



T ' / ~ 580 
The corresponding theoretical values of the delay are 
shown in the following table. 

Table VII. 

Ignition of heptane by air. Fan full speed. 

Ignition temperature calculated from previous results, 580° C. 

b_ 

To' 



= 9-5. 



T/T . 


T. 


*°C. 


« 1 m , ) t (theor.). 


t fcalc.). 


1-004 


582 


309 


012 


0-20 sec. 


1-01 


586 


313 


008 


0-13 


1-03 


597 


324 


0-04 


0-07 


1-07 


621 


348 


0-016 


0027 


111 


644 


371 


0-0076 


0-013 



Ignition of Gases by Sudden Compression. 105 

The theoretical values given in this table are represented 
by the dotted curve in fig. 9, the experimental results 
(Table V.) being shown by circles. The close agreement 
between experiment and theory is obvious ; it is, indeed, 
closer than could reasonably have been expected in view of 
the fact that the temperature errors must be estimated as 3 or 
4 degrees, while the cooling factors are subject to an error 
of about 5 per cent. The results can, however, leave little 
doubt of the substantial accuracy of the simple theory 
worked out above, and the temperature coefficient deduced 
must be very near the truth. It is of great interest to note 
that it is of the same order as that of chemical reactions in 
liquids at ordinary temperature ; for the reaction velocity is 
approximately doubled for a 3 per cent, rise in absolute 
temperature. 

XII. The only experimental values given in Tables III. 
and V. which seem to call for any special comment are 
those corresponding to experiments A 4 and A 8 . The 
" delays " found in these experiments were considerably 
smaller than those expected theoretically. This may be due 
to the fact that the measurement of very small delays is 
necessarily somewhat inaccurate with the apparatus used, 
since the speed of the piston falls off as the compression 
approaches its maximum. In such cases the lowest ignition 
temperature is, of course, reached before the piston reaches 
the top of compression, so that the measured " delays " 
which are measured from the time of maximum compression 
tend to be too small. But there is also a curious effect, 
which is invariably observed in these experiments on the 
self-ignition of carbon compounds, when the initial tem- 
perature is high, and the time of explosion short. It is 
always found that the explosion, though apparently sharp, 
is not complete, but that a fluffy deposit of carbon is thrown 
down. This deposition of carbon in an explosion has often 
been noticed by other workers when ignition is effected by 
a spark, but it is usually thought to be a consequence of 
having too little oxygen for complete combustion; in our 
experiments, however, the oxygen was always in considerable 
excess. When the minimum ignition temperature is not 
greatly exceeded, and when therefore the explosion is 
comparatively slow, combustion is complete, and no carbon 
deposit is formed. At higher initial temperature, however, 
one cannot escape the conclusion that the hydrogen is burnt 
preferentially to the carbon, and that the rate of combination 
of carbon atoms can be greater than the rate of combination 



106 Messrs. H. T. Tizard and D. R. Pye on the 

o£ carbon with oxygen. The exact conditions when this 
occurs seems well worthy o£ further investigation. 

It is always necessary to clean out the cylinder carefully 
after such a deposit has been formed, and before the next 
experiment is made ; for, if not, abnormal results will be 
obtained, and the " delay " before ignition occurs will be 
very much shorter than is expected. To explain this it 
does not appear to be necessary to attribute any special 
" catalytic " activity to the carbon ; a simple physical ex- 
planation seems to be sufficient. Such a deposit is known 
to be a very bad conductor of heat. If left on the walls of 
the piston and cylinder, we shall therefore have, on the next 
compression, large portions of gas from which the heat 
cannot get away quickly. Hence the ignition temperature 
is lowered, and the explosion takes place more rapidly. 

XIII. Experiments on the self- ignition of mixtures of the 
vapour of ethyl ether C 2 H 5 .O.C 2 H 5 gave very similar 
results. The results of the experiments on the compression 
of non-turbulent mixtures are shown in Table VIII. ; while 

Tuble VIII. 
Ignition of mixtures of ether and air. Fan stationary. 
Initial pressure (atmospheric) =I4'77 lb./sq. in. 
Mean apparent value of "7" = 1*309. 
Strength of mixture = 1 part ether to 15 of air by weight. 



No. of 


Compn. 


Initial 


Max. Compn. 


Max. temp. 


Delay 


Cooling 


expt. 


ratio. 


temp. 


pressure. 


7 = 1-309. 


obs. 


-a, 


Oi 


451 


25° C. 


105 lb./sq. in. 


201° C. 


No. ign. 


0-47 


c 2 


4-83 


24 


116 


211 


,, 


0-47 


°3 


502 


23 


121-5 „ 


214 


041 


— 


O! 


5-21 


23 


128 „ 


220 


030 


— 


C 5 


5-42 


23 


135 - 5 ,, 


226 


0-205 


— 


C 6 


5-63 


23 


143 


232 


012 


. — 


c~ 


5-82 


23 


149 


237 


0093 


— 


Cs 


6-03 


23 


155 


242-5 


0071 


— 


c 9 


0-56 


23 


172 


256 


035 


— 



Table X. shows the results when the fan was rotating at 
full speed. No ignition was observed in experiments Qi 
and C 2 in the first series, and in experiments D l5 D 2 , and D 7 
in the second series. The details of the calculation of the 
cooling factor corresponding to experiment C 2 are shown in 
Table IX. The cooling curve was not so regular in the case 
of Cj, but the mean value of the constant was the same, 




Ignition of Gases by Sudden Compression, 

Table IX. 

( 1 alculation of cooling- factor corresponding to C 2 . 



107 



Time from 
maximum 
pressure. 


Observed 
pressure. 


P-P r 


, ■*• max. — •*- f j 


•=2-8 a/*. 


00 


116-2 


44-9 










0-232 sec. 


111-7 


406 


•044 




•44 


•465 ., 


107-3 


36-0 


•096 




•47 


•70 „ 


103- > 


323 


•143 




•47 


•93 „ 


1006 


29-3 


•185 




•46 


1-16 „ 


96-9 


256 


•244 




•48 


1-39 „ 


94-7 


23-4 


•283 




•47 


1-63 „ 


92-5 


21-2 


•326 




•46 

i 


p / 


=4-83x14 


77=71-3. 




Mear 


= 0-47 







Table X. 

Ignition of mixtures of ether and air. Fan full speed. 

Initial pressure (atmospheric) = 14*77 lb. /sq. in. 
Mean apparent value of y=1'308. 
Strength of mixture as in Table VIII. 



No. of 


Compn. 


Initial 


Max. Couip. 


Max. temp, 
calc. from 


Delay 


Cooling 
factor 


expt. 


ratio. 


temp. 


pressure. 


y = 1'308. 


obs. 


=«. 


Di 


5-42 


24° C. 


135 


226-5° C. 


No. ign. 


1-33 


D 2 


573 


24 


146-5 


235 




1-28 


D 3 


6-03 


24 


155-5 


243 5 


o-io 


— 


vl 


633 


24 


164 


251 


047 




D 5 


6-63 


24 


174 


259 


0-038 


— 


Dc 


7-02 


24 


186 


268-5 


0-026 




Dt 


5-82 


22 


149 


235 


No. igu. 


1-25 


D fl 


5-94 


22 


150-5 


238 


013 


— 


D g 


6-03 


22 


155 


240 


0-087 


_ 


Dio 


6-13 


22 


161 


243 


0-078 


— 












Mean 


=1-29 



namely a =0*47. Table XI. shows the details of the cal- 
culations for D 2 ; it will be noticed that the cooling factor 
appears to diminish fairly steadily as the time increases. 
T>i and D 7 showed a similar effect : in each case the mean 
has been taken, and the mean value for the three experi- 
ments, namely a = 1'29, is probably fairly accurate. The 



108 Messrs. H. T. Tizard and D. R. Pye on the 

values for the cooling factors so obtained are lower than 
those found in the heptane experiments ; this would be 
expected, for, although the specific heats of the mixtures are 
about the same, the compression ratios used were lower in 
the case of ether mixture than in the case of heptane, since 
ether has a considerably lower ignition temperature. 

Table XI. 
Calculation of cooling factor corresponding to D 2 . 



Time from 
maximum 
pressure. 


Observed 
pressure. 


P - P /- 


p p 

, x max. *- f * 

log r-p, - =e - 


i 
a =2-3 S/t.\ 





146-5 lb./sq. in. 


61-8 








•232 sec. 


129-5 


44-8 


•140 . 


1-39 


•465 „ 


118-4 


337 


•263 


1-30 


•70 „ 


109-5 


25-8 


•379 


1-25 


•93 „ 


103-6 


19-9 


•492 


1-22 


1*16 „ 


97-7 


130 


•677 


1-34 


1'39 „ 


96-2 


11-5 


•730 


1-21 


P f =5'73x 14-77=84-7 lb./sq. in. Mean = 1-28 



XIV. Taking the ignition temperature T (without fan) as 
212° C. = 485 abs., the experimental observations of the delay 
before ignition at higher temperatures are reproduced closely 

if the value of ^ =11. In this case, we have 



T -e _._ /212-23\ 



: 0-183. 



The theoretical values for "t" calculated according to 
equation (11), are shown in the following table. 

Table XII. 



T/T . 


T. 


| 
t°C. 


1-004 


487 


214 


1-01 


490 


217 


1-03 


499-5 


226-5 


1-07 


519 


246 


1-11 


538-5 


265-5 


127 


616 


343 



a(i^)^(theor.). 



•098 

•065 

•030 

•0105 

•0046 

•0004 



t (calc.). 



)535 sec. 
•355 
•164 
•057 
•025 
•002 




Ignition of Gases by Sudden Compression. 109 

The lower curve in fig. 10 is the theoretical curve derived 
in this way ; the experimental points are shown by crosses. 

In the series of experiments with the fan running at full 
speed, "a" = 1*29, and the new ignition temperature T</ is 
approximately 235° 0.= 508 abs. 



Fig-. 10. 

















Ethyl 


ether 


and air. 








V* 












Nfcf^ 














^Nfci *~~ 








2e*pts.) 














-f- 
























1 










No ianitior 





and 



2 03 04 5 

Seconds delay. 



Hence .( W) = 1-29 x (*«) =0-54, 



Qo'_ 1-29 (235-23) _ 

Qo "0-47(212-23) °* 



., b 



Hence if fit =11, T ' should be given by 
log3-08 = 0-489 = ll(l-|j-,), 
or ^=0-956; .-. T ' = 507-5 (abs.), 

which is evidently very close to the observed value. 
Taking this value for T ', we have 

b in 485 1/w , 
T7 = 11X 507-5 =1 ° ,D - 



110 Messrs. H. T. Tizard and D. R. Pye on the 

Table XIII. gives the theoretical values for the time of 
ignition at various temperatures, calculated as already 



V11U&U ) l " 


1Y111 5 ^0 - 


J-o 
Table XIII. 




T/T . 

1-004 

101 

1-03 

1-07 

111 


T. 


t°c. 


a( m , U(theor.). 


t (calc). 


509-5 

512-5 

522-5 

543 

563-5 


236-5 

239-5 

249-5 

270' 

290-5 


0-104 

0-069 

0-0325 

0-012 

0-0055 


0-19 sec. 
0-13 „ 
06 „ 
0-022 „ 

o-oio„ 



The corresponding curve is the upper dotted curve in 
fig. 10, the experimental values being shown by circles. 
The general agreement is again all that could be desired. 
Although the ignition temperature of ether is very much 
lower than that of heptane, the temperature coefficient of 
the combustion reaction is the same within the experimental 
errors involved. 

XV. The experiments on carbon bisulphide were expected 
to be of particular interest, on account of the" anomalous 
behaviour of this substance if used as a fuel in internal 
combustion engines. It is known that for any given fuel 
the highest thermal efficiency obtainable is limited mainly by 
the tendency to " detonation " at high compression ratios. It 
is usually assumed that the tendency of any fuel to detonate 
depends upon its ignition temperature ; the lower the 
ignition temperature, the greater will be the tendency to 
detonation. According to our views, the ignition temperature 
is not a safe criterion of the tendency to detonation ; the 
temperature coefficient is also an important factor which 
must be taken into account The use of carbon bisulphide 
as a fuel illustrates this point very well ; although it has a 
lower ignition temperature than heptane, it detonates less 
easily in internal combustion engines, and not more easily, 
as might be supposed. We expected, therefore, to find, by the 
experiments described in this paper, that the temperature 
coefficient of its reaction with oxygen was very distinctly 
lower than that of heptane and similar substances. The 
experiments fully confirmed this, although the results do not 
appear to be so satisfactory in all respects as those carried 
out with heptane and ether. 




Ignition of Gases by Sudden Compression. Ill 

Tables XIV. and XV. contain all the experimental results 
of the ignition at various temperatures of mixtures of CS 2 
and air (a) when initially stagnant, and (b) when the fan was 
rotating at full speed. 

Table XIV. 

Ignition of mixtures of CS 2 and air. Fan stationary. 

Initial pressure (atmospheric) =• 14*60 lb./sq. in. 

Mean apparent value of u y" = 1*332. 

Strength of mixture = 1 part CS^ to 8 parts air by weight. 



No. of 


Compn. 


Initial 


Max. Compn. 


Max. temp. 


Delay 


Cooling 


exp. 


ratio. 


temp. 


pressure. 


y = 1-332. 


obs. 


factor. 


Ei ... 


603 


48-5° C. 


159 lb./sq. in. 


313° C. 


00S6 




E, ... 


5-63 


47*5 


143 „ 


296 


0-115 




E 3 ... 


5-63 


42 


143 


286 


0-18 




E 4 ... 


563 


39 


H7 


281 


0-26 




Eg .. 


5-63 


36 


147 


275-5 


0-35 




E(j ... 


5-63 


32 


145 


268-5 


0-42 




1 E- ... 


5-02 


32 


1265 „ 


248 


No ign. 


0-46 


E* ... 


5-21 


315 


1325 „ 


254 


0-71 




E 9 ... 


5-42 


31-5 


140-5 „ 


261 


0-59 




Eie... 


5-82 


31-5 


154-5 „ 


273-5 


034 




En- 


7-02 


31 


193 


308 


0-087 





Table XV. 

Ignition of mixtures of CS 2 and air. Fan full speed. 

Initial pressure (atmospheric) = 14V6 lb./sq. in. expts. Fi-F 6 . 



= 14-0 

Mean apparent value of 7 = 1*323. 
Strength of mixture as before. 



Fr-F, 



No. of 


Compn. 


Initial 


expt. 


ratio. 


temp. , 


Ei ... 


633 


43°C. 


F 2 ... 


6-44 


43 


F 3 ... 


6-56 


43 


:f, ... 


663 


42-5 


F« ... 


6-84 


43 


F 6 ... 


7*55 


43 


F 7 ... 


6-56 


43 


F, ... 


7-02 


43 


*Fn ... 


8-02 


43 


F 10 ... 


10-0 


42 



Max. Compn. 
pressure. 



169 lb 
171 
177 
178 
185 
208 
175 
191 
224 
? 



/sq.in. 



Max. temp. 
calc. from 
7=1-323. 



300° 

304 

307 

309 

315 

334 

307 

320 

346 

391 



Delay 


Cooling 


obs. 


factor. 


No ign. 


i-3o ; 


014 


— 


Noign.(?) 


1-38 


0-13 


1 


012 


— 


005 


— 


011 


— 


0-09 


— 


003 


— 


Ignit'd bc- 





fon top of 




compn. 





Mom 



I -34 



The cooling curves were very uniform in the above cases. 



112 Messrs. H. T. Tizard and D. R. Pye on the 

XVI. It is clear, from the shape of the curve connect- 
ing time of ignition with temperature, from the results of 
experiments without the fan, and from the difference in 
ignition temperature observed with and without the fan, 
that the temperature coefficient of the reaction is low. 
If we take T , the ignition temperature without the fan, 

to be 253° 0. = 526 abs., and ^ = 7, we obtain the 



theoretical values for the time of ignition from equation (11) 

which are given in Table XVI. 

The value of o^~~ e is 0*46 x 25 ^~ 32 = 0*193. 
1 526 

Table XVI. 

T = 526abs. Z>/T = 7*0. 



T/T . 


T (abs.). 


t°G. 


a(?^)*(theor.). 


t (calc). 


1-004 


5228 


255 


0-187 


•98 sec. 


1-01 


531 


258 


0-133 


•70 „ 


103 


542 


269 


0-072 


•38 „ 


1-07 


563 


290 


036 


•19 „ 


111 


584 


311 


0-020 


•105 „ 


1-27 


66S 


395 


0-004 


021 „ 



Fig. 11. 







1 
















it 




Carbon 


disulphic 


e and a 
















[N S. T.rr 


e scale ^ 


alf that 


of Figs. 


10 and 1 
No ionit 


"■] 








+ 




^ + 














+ 




















No 


ignition. 





0-2 0-3 04 0-5 0-6 07 OS 09 
Seconds delay. 



The theoretical values are shown by the lower curve in 
no-. 11. 




Ignition of Gases by Sudden Compression. 113 

The cooling factor when the fan is used is 1*34. The ratio 
of the loss of heat (or reaction velocity) at the ignition 
temperatures with and without fan is consequently 

Qo' 1-34(296-42) 

Qo ~ 046(253-32) 

= 3-35. 

Hence the new ignition temperature T ' should be given by 
the expression 

log3-35=0-525=£/T (l- ^ 



■(•-&)■ 



or 



^ =1-0-075 

= 0-925, 

T ' = 568-5 = 295°'5C. 

The new value of rfr7 corresponding to this is therefore 
J-o 



v-'-(«w-** 



The calculation of the corresponding delay curve is shown 

° T , I being 

l-34xg=06. 



T/T . 



1004 

101 

1-03 

107 

1-11 



571 
574 

585-5 
6085 
631 



Table XVII. 



t°c. 



298 
301 
312-5 
335-5 

358 



a(^rr^(theor.). 



0205 
015 

0-08 

0042 

0-025 



t (calc). 



'34 sec. 
•25 „ 
•13 „ 
•07 „ 
•04 „ 



This is the dotted curve shown in fig. 11, the experimental 
points being marked as before. The agreement between 
Phil May. S. 6. Vol. 44. No. 259. July 1922. I 



114 Messrs. H. T. Tizard and D. R. Pye on the 

theory and experiment is in this case only moderate. In 
particular, it will be noticed that although the ignition 
temperature with the fan running is calculated, from the 
results without the fan, to be 295°*5 C, actually no ignition 
was observed at 300°C, and even in one experiment at 307° C, 
although this is extremely doubtful, since in two other 
experiments ignition was observed to take place at 307° C. 
and 304° C, with the comparatively short delays of O'll and 
0'14 second respectively. It is possible that when the loss 
of heat is considerable, and the temperature coefficient small, 
there is an appreciable error introduced in neglecting 
changes of concentration when calculating the time of ignition 
at temperatures near the ignition temperature. This would 
account for no ignition being observed when a long delay 
was expected. The value of 6/T given above cannot, bow- 
ever, be very far wrong. For suppose we take the ignition 
temperature T ' of the mixture when, the fan is running as 
300° C. instead of 295 0, 5, and calculate the temperature co- 
efficient solely from the difference in T and T ' without 
regard to the " delay " curve when the fan is stationary ; 
then we shall have 

Ti=||6 =0 .9i8 

lo 573 

and Qo'_ 1-34( 300-42) _ 

Qo ~ 0-46 (253 - 32) ~ 6 W ' 

log3-40 = -531=A(l-g,) 
= |- x-082; 

J-O 

— =6*5 instead of 7*0. 

J-o 

This value for ~r would, in fact, fit the lower part of the 

J-o 

delay curve without the fan rather better than the value 7*0, 
but the calculated delay curve when the fan is running- 
would then be some way from the experimental points. If 
we take a mean value 

6/T =6-7, 
we shall be very unlikely to be as much as 10 per cent, from 




Ignition of Gases by Sudden Compression. 115 

the true value, even when all possible sources of error are 
taken fully into account. 

XVII. It has already been mentioned that when the time 
of ignition by compression of hydrocarbons (and of ether) is 
small, i. e. when the gases are suddenly compressed to a 
temperature well above the ignition temperature, carbon is 
thrown down, even though excess of oxygen is present. In 
the experiments with CS 2 an even more curious phenomenon 
was noticed. In this case, whereas the sulphur burns to 
S0 2 when the initial temperature of compression does not 
exceed very greatly the lowest ignition temperature, when 
the initial temperature is high the products of combustion 
smell strongly of JBLS. For instance, the products of com- 
bustion in experiments E x and E n above both smelt strongly 
of H 2 S, although in E 2 -E ]0 inclusive only S0 2 could be 
detected by smell. It was also possible to detect H 2 S after 
experiments F 6 and F 10 (with the fan), the smell being 
particularly strong in the case of F 10 . The smell of H 2 S 
could also just be detected along with S0 2 in experiment F 9 , 
whereas in the remainder only S0 2 could be detected. 

The H 2 S could only have come by combination with w r ater- 
vapour present in the air, which was not dried. This 
occurrence of H 2 S is all the more interesting since it is 
known that a perfectly dry mixture of CS 2 and oxygen can 
be exploded by a spark, whereas perfectly dry mixtures of 
other gases, e. g. carbon monoxide with oxygen, cannot. It 
is possible that some such reaction as 

CS 3 + 2H s O = CO s + 2H s S 

takes place, followed by the combustion of H 2 S ; but even if 
this is the case, it would be expected that the H 2 S would be 
quickly burnt in presence of excess of oxygen under the 
conditions of these experiments. Further experiments on 
the ignition of H 2 S itself will probably throw some light on 
these observations. 

XVIII. In Table XVIII. are summarized the chief results 
of the experiments described above. The ignition tempera- 
tures represent the lowest average temperatures at which 
the non-turbulent mixture could be caused to ignite. The 
rates of evolution of heat at these temperatures for the three 
cases are calculated from the cooling factors and the specific 
heats of the mixtures. 

12 



116 Messrs. IL T. Tizard and D. R. Pye on the 

Table XVIII. 





Heptane 
C 7 H 16 . 


Ether 
C 2 H 5 .O.C 2 H 5 . 


Carbon 
bisulphide 

cs 2 . 


Composition of gas by weight... 1 : 20 of air. 

i 


1:15 


1:8 


T =igni(ion temperature " 280° 0. 


212° 0. 


253° C. 


c v at ignition temperature ' 0'20 0'20 


0-18 


Rate of evolution of beat due to; 25 calories . ~ 
reaction per gram of mixture per second. [ 


18-5 


Total beat of combustion pei 


1 510 calories. 510 calories. 


386 calories. 




Value of b/T 


■ io-o+5 % ! n-o+5°/_ 


6-7 ±10% 









XIX. In order to calculate the true temperature co- 
efficient B (see equation 5) from the values of b/T , it is 
necessary to examine the significance of T a little more 
closely. As already stated, T is a measure of the lowest 
average temperature of the gas at which ignition takes 
place. Now the actual temperature of the gas after sudden 
compression can hardly be uniform throughout ; in fact, 
when the gas ignites after a considerable delay, it is always 
found that the pressure, and therefore the average temper- 
ature, falls, in some cases quite considerably, before ignition 
takes place throughout the mass. This shows clearly that 
that portion of the gas which ignites at first has initially 
a higher temperature than the average, thus confirming 
Dixon's experiments. Absence of information as to the 
temperature gradients which may exist under these conditions 
has no doubt led Nernst and Dixon in their experiments to 
calculate the ignition temperature as if the compression were 
adiabatic, and to ignore the influence of loss of heat during 
compression and before ignition. They assume, in fact, that 
that portion of the gas which does ignite is at the adiabatic 
temperature. 

It is hardly likely, however, that big differences in 
temperature exist after compression when the gases are in a 
turbulent state ; and the fact that the temperature coefficients, 
calculated from the differences in "average" ignition 
temperature between turbulent and non-turbulent mixture, 



Ignition of Gases by Sudden Compression. 117 

agree so well with those calculated from measurements of 
time of ignition at various temperatures with non-turbulent 
mixture, confirms the views taken in the previous paper 
(Tizard, loc. cit.) that it is unlikely also that any big 
differences of temperature exist in the non-turbulent mixture 
after compression. In the absence of direct evidence on 
this point, however, it is important to calculate the " adia- 
batic " temperatures also in the above cases. 

The mean specific heats C„ per gram molecule are : 

Heptane (room temperature-300°C.) = 50 calories 

approx. 
Ether (65°-230° C.) = 33'6 calories (Regnault), 
CS 2 (70°-194° O.)=10'0 (Regnault). 

Taking these values, and C y = 5'0 for air, we obtain the 
figures for the mean true value of " y" and the corresponding 
adiabatic temperatures, given in the following table. 



Table XIX. 



Heptane. 


Ether. 


OS 2 . 




Mean apparent value of 

" y " (observed) 

Mean true value of y ... 
" Average" ignition teni- 


1-313 
1-353 

553 abs. 


1-309 
1-347 

485 abs. 

516 

31°C.=6-4°/ 


1-332 
1-384 

526 abs. 

572 

46° 0. =8-7% 


il Adiabatic" ignition 


591 

41° C. =7-4% 







The average specific heat for CJS 2 taken in the above 
calculations is probably too low, since it refers only to a 
range of temperature up to 194° C, whereas the ignition 
temperature was 260° C. 

It will be observed that the difference between the average 
observed and the theoretical adiabatic temperatures is not 
very great ; we consider that the "average" temperature is 
probably closer to the true ignition temperature than is the 
"adiabatic" temperature, but for the purpose of estimating 
every possible source of error in the temperature coefficients, 
it is better at this stage to recognize the uncertainty, and 
take for the true values of the ignition temperatures the 
values 

Heptane 573° absolute 

Ether 500° „ V ±4 per cent. 

CS 9 549° „ 



} 



118 Messrs. H. T. Tizard and D. R. Pye on the 

Hence, from the values of =^ we get finally for the true 
temperature coefficient B : ° 

Temperature coefficient B 
(equation 5). 

Heptane-air... 13,200+ 9 per cent. 
Ether- „ ... 12,600 ± 9 „ 
0S 2 - „... 8,500 + 14 „ 

The significance of these figures will perhaps be better 
appreciated by the statement, that the percentage rise in 
absolute temperature necessary to treble the reaction velocity 
is 4 per cent, in the case of heptane and ether and 7 per 
cent, in the case of CS 2 . 

XX. Of recent years, considerable attention has been 
directed to the "radiation" theory of chemical reactions. 
According to this theory, the ultimate cause of any chemical 
reaction is to be found in the absorption of radiation of a 
frequency which depends upon the nature of the reactants. 
In the case of the majority of chemical reactions, namely 
those which are not " photochemical " in nature, this radiation 
will belong either to the visible, or more usually in the short 
infra-red part of the spectrum. The supporters of the theory 
hold the view that it is only through the absorption of such 
radiation that a molecule is able to acquire that excess of 
energy, over the average at any temperature, which enables 
it to decompose or to react with another molecule. The 
frequency of the radiation is therefore known as the 
"activating" frequency. 

This reasoning leads to the conclusion that the temperature 
coefficient B of a mono-molecular reaction is determined by 
the relation 

B-£, d2) 

where "»>" is the activating frequency, which should 
correspond to an absorption band in the reacting species. 
No reliable experimental evidence has yet been brought 
forward in support of this theory, but in view of the scanti- 
ness of the data existing on homogeneous gas reactions, it is 
of particular interest to apply it to the results of the 
experiments described above. 

In attempting to apply the theory, a difficulty at once 
arises. Evidence has been brought forward to show that 
the ignition temperature of substances with oxygen is 



Ignition of Gases by Sudden Compression. 119 

practically independent of the concentration of the com- 
bustible substance. If the rate of the reaction were deter- 
mined solely by the amount of oxygen present, we might 
expect the temperature coefficient also to depend solely on 
the oxygen, and therefore to be the same in all cases. This 
is clearly not true. Nor does the temperature coefficient, in 
the case of the heptane explosion which has been most closely 
investigated, correspond at all closely to that calculated by 
means of equation (12) from the infra-red absorption of 
oxygen. Oxygen has an absorption band corresponding to 

\ = &2fi, or j/ = '94x10 14 ; hence, since \ =4*86 x 10" 11 , 
we should have * 

B = 4550 (calc.) instead of 13,200 + 9 per cent. (obs.). 
It is clear that equation (12) cannot be applied. On the 
other hand, if the rate of reaction depended on the product 
of the number of active molecules, both of oxygen and the 
other reactant, we should expect, on the same theory, to 
find the temperature coefficient given by 

B = yO'i + 'a), 

where v u v 2 correspond to absorption bands in the reacting 
substances. 

Now, all hydrocarbons have a weak absorption band at 
\ = 24 /x, and a fairly strong one at 3*43 /jl. Taking 
\ = 2*4 fi, which is most favourable to the theory, we have 
v 2 = 1-25 x 10 14 and Vl (oxygen) ='94 x 10 14 . 

Hence B = 4-86 x 10" 11 x 2'2 x 10 14 = 10,700. 

This approaches more closely the experimental value 
B = 13,200 ±9 percent. 
It must be pointed out, however, that this approximate 
agreement is only obtained by an assumption as to the actual 
mechanism of the reaction which does not agree with the 
existing experimental results. 

The failure of the " radiation theory " to account for the 
results obtained in these experiments is more significant 
when we regard it in a different way. The theory requires 
that the rate of a chemical reaction should be proportional 
to the density in the reacting system of the radiation which 
is absorbed by the reacting substances. Now, in the case 
of gases which are caused to react by a rise in temperature 
due to sudden compression, the radiation density must 
remain practically unchanged, for the temperature of the 
walls remains constant. It may be momentarily increased 



120 Messrs. H. T. Tizard and D. R. Pye on the 

owing to the sudden compression, but such an increase 
cannot persist during the period of delay, and in any case is 
negligible compared with the increase in radiation density 
which would occur if the temperature of the walls of the 
vessel were raised to the compression temperature of the 
gas. Again, the emission and absorption of radiation by 
the gas itself at the compression temperature of 500-600 
absolute is negligible compared with that of the solid walls. 
Hence we arrive at the conclusion that, although the density 
of radiation in the system is not appreciably changed, the 
gases react at a high rate. This fact appears to us to prove 
conclusively that the radiation theory cannot be accepted 
either in its original form, or as modified to meet its failure 
to account quantitatively for the temperature coefficients of 
chemical reactions in liquids under steady conditions of 
temperature. 

It must be pointed out, however, that in spite of this con- 
clusion, there does seem to be an indirect connexion between 
the effect of temperature on the rate of combustion of many 
substances and their absorption of infra-red radiation. For 
example, Ooblentz has shown that all paraffin hydrocarbons 
have very similar absorption spectra, with a weak band at 
\ = about 2*4 //,, and strong bands at X = 3*43, 6*86 yu,, etc. 
Now, Bicardo has shown that the tendency of hydrocarbon 
fuels to detonate in an internal combustion engine depends 
consistently on their ignition temperatures as determined in 
the manner described above. According to our views this 
is strong evidence that the temperature coefficients are 
practically the same throughout. Again, it has been shown 
that ethyl ether has approximately the same temperature 
coefficient of combustion as that of heptane; while Coblentz 
has found that it's absorption spectrum is also nearly 
identical, with bands at 2*4 p and 3'45 /x. Carbon bisulphide, 
on the other hand, has a much smaller tendency to detonate 
in an internal combustion engine than heptane, although it 
has a lower ignition temperature; corresponding to this we 
find that the temperature coefficient is low, and that the first 
strong absorption band in the infra-red occurs at A, = 4* 6 //,. 
Finally, hydrogen " detonates " easily in spite of its high 
ignition temperature ; its temperature coefficient must 
therefore also be high, a deduction which is confirmed by 
some preliminary experiments we have made on the delay 
before the ignition of a non-turbulent mixture of hydrogen 
and air. We should expect from this point of view to find 
an absorption band in the short infra-red region (say about 
1*0 yu,); actually no absorption is observed, but that the 



Ignition of Gases by Sudden Compression. 121 

frequency of atomic oscillation is high is in agreement with 
our general knowledge of the hydrogen molecule. In spite, 
therefore, of the strong arguments that have been brought 
forward against the radiation theory of chemical reactions, 
these results support the view that there is a connexion, 
even though an indirect one, between the temperature co- 
efficients of gaseous reactions and the infra-red spectra of 
the reacting substances. 

XXI. The results of this investigation may be summarized 
as follows : — 

(a) Quantitative experiments confirm the view that at the 
lowest ignition temperature the heat evolved by the 
combustion of a gas just exceeds that lost to the 
surroundings. 

(6) From measurements of the rate of loss of heat just 
below the ignition temperature, and of the intervals 
between the end of compression and the occurrence 
of ignition at different temperatures, it is possible to 
deduce the temperature coefficient of the gaseous 
reaction. 

(c) The temperature coefficients so obtained are confirmed 

by the increase in the minimum ignition temperature 
which is observed when the gas is in a turbulent 
state. 

(d) The results show that the temperature coefficient of 

the combustion of carbon bisulphide is much lower 
than that of heptane or ether. This is in agreement 
with the relative tendencies of these fuels to detonate 
in an internal combustion engine. 

(e) The results do not agree with the radiation theory of 

chemical reaction. 
(/) Some evidence is put forward to show that the rate 
of reaction on sudden compression is independent 
within wide limits of the concentration of the com- 
bustible gas, but only depends on the amount of 
oxygen present. This evidence is, however, incom- 
plete. 

We are greatly indebted to Messrs. Ricardo & Co. for the 
loan of their apparatus and for much additional assistance ; 
also to the Department of Scientific and Industrial Research 
for a grant towards the expenses of the investigation. We 
also take this opportunity of thanking Mr. C. T. Travers for 
his help in carrying out some of the experiments. 



[ 122 ] 

IX. On the Vibration and Critical Speeds of Rotors. 
By C. Ropgeks, O.B.E., B.Sc, B.Eng., M.I.EE* 

NUMEROUS papers have been written on the question 
of the whirling and vibration of loaded shafts and 
kindred subjects, and the calculation of the first critical 
speed — the lowest speed at which the vibration shows a 
maximum value, is now a matter of daily routine in 
designing offices. 

This critical speed can be calculated with sufficient accuracy 
for practical purposes and as a rule the running at speeds 
not in the neighbourhood of that indicated by the calculation 
is free from vibration. But cases occasionally arise where 
troubles from vibration occur at speeds above or below the 
calculated critical speed, the reason for which is obscure and 
the remedy correspondingly difficult to find. 

It is the object of this paper to discuss various subsidiary 
causes which might conceivably lead to unsatisfactory run- 
ning at other than the usual calculated critical speed, but 
while these are indicated as possible causes of disturbance, 
it is not to be assumed that these causes always exist or that 
they will always induce disturbed running. The object is 
rather to indicate reasons why vibration might possibly arise 
and thus if an actual case occurs, to suggest a clue to the 
cause. 

Although the basis of the paper is a physical or mechanical 
one, the treatment is largely mathematical, as it is only 
by this means that formulae can be obtained from which 
numerical results can be worked out. 

The phenomena when a rotor vibrates are complicated, as 
the shaft is supported in the bearings on a film of oil, the 
thickness of which is continually changing, the bearings and 
foundations are not themselves perfectly rigid, and there is 
a certain amount of initial bending of the shaft (and to a 
much smaller extent of the rotor body) due to gravity. If 
the rotor consists of a number of disks as in a steam turbine, 
there is also the inter-action of the forces of each disk on 
the others. 

For the sake of simplicity, we shall confine our attention 
to the case of a single part rotor, either a disk or a cylinder, 
rigid as regards bending and mounted on an elastic shaft 
running in rigid bearings. Some effects of non-rigidity of 
the rotor and bearings and of alterations in the thickness 
of the oil film in the bearings will be indicated. 

* Communicated by the Author. 



On the Vibration and Critical Speeds of Rotors. 123 

A single part rotor can vibrate in either of two ways, as 
shown in figures 1 and 2, or in a manner which is a combi- 
nation of the two motions : — 

Fig. 1. Fig-. 2. 







Fig. 1 shows a purely transverse vibration, while in fig. 2 
the motion is solely one of oscillation about the centre of 
gravity. In the transverse vibration the conditions are 
clearly the same whether the rotor body is a disk or is 
cylindrical ; but in the case of the oscillation, the motion, 
owing to the gyrostatic effects called into play, depends 
both on the proportions of the rotor and the speed at which 
it is running. 

The speed at which transverse vibration becomes a maxi- 
mum we shall call the "first critical speed/' and that at 
which the oscillation becomes a maximum, the "second 
critical speed," as the latter is in all practical cases con- 
siderably higher than the former. 

The following is a general outline of the treatment adopted 
and the conclusions reached : — 

Section 1 deals with the vibration of a rotor when not 
running, and a relationship is deduced between the fre- 
quencies for the transverse motion and for the oscillation 
which we shall call respectively the " stationary first critical 
speed," and the " stationary second critical speeds." 

The second section deals with the transverse vibration, 
frictional resistance being ignored. It is first showm that 
there appears to be no foundation for the frequently ex- 
pressed view that there is a possible region of marked 

vibration at — ^ times the first critical speed, as such a 

conclusion can only be reached through an incorrect assump- 
tion with regard to the conditions. It is then shown that 
the motion or vibration is a circular whirl about the statically 
deflected position of the shaft, and that this motion reaches 
a maximum at a speed equal to the stationary first critical 
speed. The magnitude of the whirl is proportional to the 
amount by w'hich the machine is out of balance, so that 
the main vibration here dealt with should disappear with 
good balancing. 

The action of gravity is then gone into more fully, and it 



124 Mr. 0. Rodger s on the Vibration 

is shown that in addition to producing the ordinary static 
deflexion, the action o£ gravity is such as to cause a double 
frequency ripple in the whirl which would tend to reach a 
maximum at half the first critical speed. The magnitude of 
this ripple is, however, proportional to the square of the 
amount by which the rotor is out of balance, and would 
therefore fail to appear in a well-balanced machine. In any 
case the effect is very small. 

It is then shown that a rotor with bi-polar asymmetry, 
such as exists in a rotor slotted for a two-pole winding, may 
show a double frequency vibration at half the critical speed 
even when the rotor is perfectly balanced, so that such a 
machine might vibrate at half the critical speed even when 
it would run perfectly at the full critical speed. Vibration 
arising from this cause could not, therefore, be rectified by 
balancing, and this is the only case met with where improved 
balancing would not effect an improvement in the running. 
This case is gone into in some detail, and it is shown that 
the motion here also is a circular whirl of double frequency, 
that is, of twice the speed of rotation of the machine. If, in 
addition, the machine is out of balance, a triple frequency 
effect might appear, but is not likely to do so. 

The effect is then discussed of lack of proportionality in 
the deflexion of the shaft and again the possibility appears 
of vibration appearing at half the critical speed, but only if 
the machine is not properly balanced. The effect is then 
gone into of fluctuations in the angular velocity through 
variations in the driving torque, and of resonance between 
the rotor and the foundations or other masses outside the 
machine, from which it appears that marked vibration might 
appear at almost any speed through either of these causes. 

The effect of friction on ihe transverse vibration is then 
discussed, and the results are given for the case where the 
frictional resistance varies as the first power of the speed, 
and also where it varies as the second power of the speed, 
the latter being more probably in accordance with the facts 
than the former. It is shown that the maximum vibration 
appears in both cases at a speed equal to the stationary 
critical speed, also that the phase difference between the 
force due to the out-of-balance and the displacement 
depends on the amount of friction, and also on the speed. 
If the frictional forces vary as the square of the speed, as 
is probably the case, the angle varies also with the amount 
by which the rotor is out-of-balance. 

Some effects of viscosity of oil in the bearings, and of 
different bearing clearances are then gone into. 



7?r 



and Critical Speeds of Rotors. 125 

In Section 3, the oscillatory vibration is dealt with, 
taking into account the gyrostatic effects when the machine 
is rotating, but ignoring the friction in order to keep the 
expressions as simple as possible. It is there shown that 
the gyrostatic effect causes the point of marked vibration to 
occur at a higher speed than would be the case if the machine 
were not rotating, and simple rules are given for calculating 
this vibrating speed. An example is added to illustrate the 
method of working the rules given. 

Much of the work on the main transverse vibration and 
the main oscillatory vibration has been dealt with in various 
forms by Ghree, Stodola, Morley and others, and the solution 
for the transverse vibration with friction depending on the 
first power of the speed has been given by H. H. Jeffcott 
(Phil. Mag. March 1919), but the ground covered by the 
remainder of the paper, particularly the question of sub- 
sidiary critical speeds, does not appear to have received 
much attention ; there is, however, in 'Engineering' a dis- 
cussion where subsidiary critical speeds are touched on, 
arising out of a paper by W. Kerr in that journal 
(Feb. 18th, 1916). 



Section I. — Stationary Vibrations. 
A. Transverse Vibrations. 

1. If M is the mass of the rotor body (the mass of the 
shaft being being neglected), and we assume the rotor to be 
perfectly balanced, the shaft will, when not rotating, show a 
deflexion measured at the centre of gravity of the rotor of 

where a is the force required to produce unit deflexion. The 
method of working out the static deflexion of the rotor for 
actual cases is well understood and the value of a can be got 
from the deflexion diagram. 

2. If now the rotor is set in vibration in a vertical plane, 
the motion is represented by the following equation (using 

d 2 y 
fluxional notation, where y is written for -~^ and y for 
7 at 

^,etc.) 

■ ' My + <ry + M ff = (2) 



126 Mr. 0. Rodgers on the Vibration 

The solution is 

^MVl^y^ ■ ■ ■ m 

where Nx and y ± are constants the values o£ which depend 
on the initial conditions. The vibration therefore takes 
place about the statically deflected position as a centre, and 

with a frequency of vibration of — c l9 where 

c '=Vm w 

This vibration takes place in a vertical plane and may be 
considered as the resultant of two vectors rotating in oppo- 
site directions, each with an angular velocity of \/ ^ . If 

o- and M are expressed in eg s. or f.p.s. units, this angular 
velocity will be in radians per second and since from (1), 

^ is numerically equal to -^ , the speed of either of these 
vectors in revs, per minute will be _: — A / — . If, further 

2-7T V y Q 

g and y are in c.g.s. units we have the formula 

WTJ ,, 60 /Ml 300 . , . /K . 

where y is the static deflexion in cm. 

3. It will be seen afterwards that, as is well known, this 
formula gives the first critical speed in R.P.M. ; this is to 
be expected, as the out-of-balance forces will then resonate 
with the natural free vibrations, with the result that the 
latter will become of considerable magnitude. 

B. Oscillatory Vibrations. 

1. If the rotor is twisted about its centre of gravity so 
that the deflexion is in a vertical plane, and is allowed to 
oscillate freely, the motion is represented by 

B^ + /c^ = 0, (6) 

where B is the cross moment of inertia, that is, the moment 
of inertia of the rotor about a line through the centre of 
gravity at right angles to the shaft, yjr is the angle through 



and Critical Speeds of Rotors. 127 

which the axis of the rotor at its centre of gravity is 
deflected from the stationary position, and k is the torque 
required to produce unit angular deflexion. 
The solution is 

^ = N 2 sin(^/^- 72 ^ .... (7) 

where N 2 and y 2 are constants the values of which depend 
on the initial conditions. The frequency of the oscillation 
is therefore 

iiv b or L c " where 

'-a/b («) 

As already indicated, we cannot at once deduce from this 
what will be the actual second critical speed, owing to the 
gyrostatic effects, but the result is of importance, as it 
simplifies the calculation of the actual second critical speed, 
as will be shown later. We shall in what follows call c 2 the 
stationary second critical speed. 

2. It should be pointed out that there is a simple relation 
between c x and c 2 which greatly facilitates the calculation of 
the stationary second critical speed in those cases where the 
centre of gravity of the rotor is midway between the bearings. 
If 21 is the distance between the bearing centres and P 2 the 
force exerted by the deflected shaft on either bearing, 

/n/r = 2P 2 Z. 

The angle y]r is very small so that the force P 2 is the same as 
would be required to depress the shaft through a distance 
-tyl if the rotor were held rigidly. Now we have seen that 
the force M</ at the centre of gravity causes a transverse 

deflexion of ?/ = , and as yfr is small, 

so that Mg — ayfrl, 

also /n/r = 2P 2 Z and F 2 = ±Mg ; 

therefore fcyjr = MpZ, 

so that K = a-l 2 (9) 

and (8) becomes 7 / a .■ 

C2==/ V B' ^ ' 



128 Mr, 0. Kodgers on the Vibration 

and if B = M^ 2 2 , 



c ^h 



\/w ~N 



(12) 



and comparing with (4) we thus get : 

C\ Aug 

c 2 ~~ I 

We thus find that 

First critical speed (transverse vibration) 
Stationary Second critical speed (oscillation) 

_ Radius of Gryration for the cross moment of inertia 
Half the distance between the bearing centres 

This is a useful formula for calculating the stationary 
second critical speed when the first is known, for cases 
where the centre of gravity is midway between the 
bearings. 

It shows that with cylindrical rotors of this type the 
second critical speed must always be considerably above 
the first, and the only instance in normal designs in which 
the second critical speed could be lower than the first would 
be that of a flywheel mounted on a short shaft. 



Section II. — Transverse Vibrations — First Critical 

Speed. 

A. Neglecting Junctional Resistance. 

1. It will simplify the treatment of this question if we 
first consider the case of a rotor unimpeded by frictional 
resistances set up by the air and then treat separately the 
effects produced by friction. 

The conditions obtaining when a rotor is not perfectly 
balanced and is rotating are illustrated in fig. 3, where 
represents the position of the centre line c f the bearings, 
and C the deflected position of the centre line of the shaft, 
while Gr shows the position of the centre of gravity of the 
rotor. thus gives the undeflected position of the shaft 
centre line and 00 = r the shaft deflexion at any instant, 
while 0Gr = <? is the error in the centering of the rotor; 
M<7 is the weight of the rotor acting vertically downwards. 

The rotation of the rotor about its centre line, i. e. the 
rotation imparted by the prime mover, is represented by 
the motion of G around 0, i. e. by the rate of change of 0. 



and Critical Speeds of Rotors. 129 

The whirling of the rotor is represented by the motion 
of C about the undisturbed position of the shaft centre line, 
i. e. by the rate of change of a. 

The " vibr.ition" of the rotor is judged in a general way 
by the vibration of the bearings as felt when the hand is 
applied to them. The force on the bearings is that applied 
along OG by the deflexion r of the shaft, and vibration of 




the bearings arises through the varying position and magni- 
tude of OC ; these in turn are due to the motion of the 
ceritre of gravity G. 

2. If the machine is steadily rotating it might at first 
sioht be thought that OC and CGr would be in the same 
straight line, so that Gr would be steadily revolving together 
with G about the undisturbed position of the shaft centre 
line with an angular velocity n say. At the same time the 
deflexion 00 = ?' might be changing its value and (neglecting 
the weight of the rotor) the motion would thus be given by 

Mr — M?i 2 (V-f e) + ar = 0, 

or putting crjM = Ci 2 , 

r + (c x 2 — n 2 ) r = n 2 e, 

the solution of which is 

where Nx and y l are constants. 

We should thus conclude that r would become unlimited 
Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. K 



130 Mr. C. Rodgers on the Vibration 

in magnitude when the angular velocity n is equal to c x or 
\f y. , and, further, that the variation in the magnitude 
of r consists of a free vibration having a periodicity of 

l /in 

2tt V c?-n 2 ' 

This result for the periodicity of the free vibration would 
lead to the conclusions that when the machine is not rotating 
(n = 0) the periodicity is c l5 the same as for the stationary 
transverse vibration, and that when running at the critical 
speed (n = ci) the periodicity is zero. There would thus be 
some intermediate speed where the periodicity of the free 
vibration corresponds with the running speed, and resonance 
might take place. This would occur when 

c 2 — n 2 = n 2 

We should thus be led to expect marked vibration when 

the running speed is — ^ x the critical speed. 

This conclusion and the argument on which it is based are, 
however, erroneous. In the first place, the assumption is 
made that co or 6 is constant and further the condition 
is omitted that, as all the forces pass through O, the angular 
velocity about must be constant, or r 2 = h, say. The 
correct equations for the free vibrations are thus : 

jV( Cl »-#)r=b,i 

r 2 = h.) 

This does not admit of direct solution *, and it is simpler 
to use rectangular co-ordinates, as we shall now proceed 
to do. 

* The solution is, however, well known and is given in books on 
Dynamics dealing- with Central Forces : — 

If p is the length of the perpendicular from the centre of force on the 
tangent to the path, it is known that 

h 2 dp 

— —■ — err, 

p 6 ar 

giving — = a constant — err 2 , 

which is the pedal equation of a central ellipse. 



and Critical Speeds of Rotors. 131 

3. With the notation given in fig. 3, it will be seen that 
th^ position of the rotor is completely defined by the co- 
ordinates x, y, and 6 {x and y being the co-ordinates of" G) 
and only three equations are required to express the motion 
fully ; the value of the whirling angle a follows from the 
magnitude of the other co-ordinates. 

The force exerted by the deflected shaft is err, the com- 
ponents of which are 

— a{x — e cos 6) along OX and 

— a(y — e sin 6) along OY. 

Resolving along OX and OY and taking moments about 
G, we thus have : 

M£ + <7(.i'-*cos0) = O, (13) 

My + (r(y-esm0) + Mg=Q, .... (14) 

M^+«ra(tfMn0-yco!«0) = O, . . . (15) 

where k l is the radius of gyration of the rotor about the 
longitudinal axis through the centre of gravity. 

In practice the rotor is driven at an average angular 
velocity co say, which will vary from constancy only by 
small amounts which we shall find later are negligible. 

Assuming as a first approximation that the angular 
velocity is constant = co, so that 6 — cot, (13) and (14) then 
become, writing cr/M = c 1 2 as before, 

x + c l 2 x = Ci 2 e cos cot, (16) 

y + c l 2 y = c] 2 e sin cot — g, .... (17) 

while (15) becomes an identity. 
The solutions are 

■jjrnN^in M-yi)-f — ^-^coscot, . . . (18) 



y = N 2 Cos ('•!* — 71J+ 2 Cl e 2 sin cot-g/c^, . (19) 

C'i — CO 

where N 1; N 2 , and y l are constants. 

4. The above results give the motion of the centre of 
gravity G. The motion of the centre of the shaft C is to be 
found in a similar way, still on the assumption that the 

K 2 



132 Mr. 0. Rodgers on the Vibration 

angular velocity of rotation is constant. If x 1 y' are the co- 
ordinates of C, 

,v = x' 4- e cos o>£, 



giving the equations 



■// = t/ -\-e sin cut 



it +Ci'V =W^COSG)^ (20) 

y' +cfy' = oo~e sin t»£ — <?, .... (21) 
the solution of which is : 

9 

x' = TX l sm{C l t-y 1 ) + -p^GQ8<ot, . . . (22) 
v' = N 2 cos (CV — 7i) h 2 — -^sin w^— ^/c*! 2 . . (23) 

('l — CO 

5. These equations are the same as for a perfectly balanced 
rotor with a weight attached to it of such small value as not 
to affect the position of the centre of gravity, or the down- 
ward pull due to gravity, but producing a force of Mco 2 e. 
In other words, we can treat the unbalanced rotor as if it 
were a perfectly balanced rotor with a force Mw 2 <? attached 
to it, and as this mode of presentation is easier to follow than 
the former, we shall employ it in the remainder of the paper. 

6. It will be seen that the solutions for the motions of the 
centre of gravity and the centre line of the shaft are the 

same, except that the former has an amplitude - 2 — 2 , and 

CO" € 

the latter an amplitude of — ^ - 2 , so that they differ by the 

amount e ; this shows that 00 and CG are in the same 
straight line when not running at the critical speed. 

7. The solutions show firstly that the motion takes place 
about the position 

Ma q 

which we have seen is the statically deflected position of the 
centre of gravity of a perfectly balanced machine. It is 
sometimes contended that as the speed increases the rotor 
shaft tends to straighten out, but there is no indication in the 
present treatment that this is the case. 

8. The free vibration 

x — N t sin {i\t — 71), 

y = N 2 cos fa*- 7O, 

is the same as for a perfectly balanced rotor and has a 



and Critical Speeds of Rotors. 133 

frequency the same as the frequency for stationary vibra- 
tions, and, as we shall presently see, ii' expressed in R.P.M., 
is the same as the first critical speed. It is, therefore, inde- 
pendent of the speed of rotation of the rotor, and there is no 

possibility of resonance occurring at -y^ X the critical speed 

as suggested by the erroneous method mentioned earlier in 
this section. The free vibration itself is thus represented by 
two components having the same frequency, but different 
amplitudes; it is therefore a central ellipse, the centre 

being at the point y= . 

It will be shown later that the free vibration is damped 
out by friction, so that it has no importance in practice. 

9. The forced vibration for the centre C of the shaft is 
given by 

x= —. — . 2 cos cot, (24) 

Ci — G) 

we 
£=_-—_ sin cot-a/cj 2 . . . . (25) 



6*i — ft) 



This naturally has the frequency corresponding to the 
angular velocity of rotation o>, and has a maximum value of 



we c , 

7. tor each axis. 



Cj — CO' 

The motion of the centre is thus a whirl, the radius or 
amplitude of which is proportional to the out-of-balance and 
is zero when the rotor is perfectly balanced. A perfectly 
balanced rotor, therefore, cannot whirl in the manner ex- 
pressed by equations (24) and (25). 1 

The amplitude of the whirl is also proportional to -j 2 

and it thus becomes a maximum when * 

(i)= +Ci. 

(The sign + merely indicates that the rotation may be in 
either direction.) 

This value of co gives the first critical speed, which is thus 
the same as the stationary frequency for transverse vibra- 
tions. Reasons will be given later why the radius of whirl 
does not become infinite at the critical speed, i. e., why the 
shaft does not break when the rotor reaches this speed. 



134 Mr. C. Rodgers on the Vibration 

10. At the critical speed where co 2 = c\ 2 equations (20) 
and (21) become : 

x + c 2 x = c'i 2 e cos C]t, 

y + cfy = c 2 e sin erf — g, 

which admit o£ a solution not involving infinite values, 
namely, 

# = £ — sin c x t 

= *f «*(«.«- |) (26) 



y=—i^ r cosc i t — (?/c l 2 



<\e . / 7r 



= iysini^-- 



)-#i 2 (27) 



Equations (26) and (27) thus give the motion at the first 
critical speed when friction is ignored. They show that the 
component along each axis has an amplitude which con- 
tinually increases in proportion to the duration of the 
motion, in other words the motion at the critical speed is 
a spiral of continually increasing radius. 

11. From (24) and (25) it will be seen that the phase 
difference of the motion with respect to cot changes from 
zero to 180° as co passes through the value c l} i. <?., as the 
speed passes through the critical ; (26) and (27) show that 
at that speed the motion lags behind cot by 90°. It will be 
seen later that when friction is taken into account, the lag- 
increases gradually as the speed increases, being still 90° 
when &> = <?!. 

12. Up to this point we have treated the angular velocity 6 
as a constant = &), as on the average it will be in practice. 
Suppose now that it varies slightly from constancy, so that 
the angular position cat becomes cot + u, where u is so small 
that its square can be neglected, and we may write smu = u 
and cos?/ = l. We then have cos 6 = coscot — u sin cot and 
sin = sin cot+u cos cot, also = u. Substituting these values 

in equations (20), (21), and (15), and writing -^ =c{ 2 as 
before, we get 

x -f Ci 2 x = w 2 e cos cot— co 2 en sin cot, . . . (28) 

y + c l 2 =y co 2 e sin cot + co 2 eu cos cot— g, , . (29) 

MM + cre(,T sin cot —y cos cot + ant cos cot + yu sin cot) = 0. (30) 



and Critical Speeds of Rotors. 135 

As a first approximation we substitute in (30) the values 
already found tor .t* and ?/, as given in (24) and (25), for the 
forced vibration and thus obtain (neglecting u in comparison 
with 1 radian) : 

2 

M/iyzV + o-tf 2 — 2 2 a = —<reg]c 2 cos cot, . . (31) 

giving for the forced vibration (the free vibration may be 
ignored as it will be damped out) : 

e(c x 2 -co 2 ) ,_ OA 

Z ' = * 2 2 27 2/ 2 2\-gCOSO)t. . . (.32) 

At the critical speed co = c^ and ^ = 0, z. 0., the angular 
velocity of rotation is constant ; at other speeds than those 
in the neighbourhood of the critical, the term co 2 e 2 c 2 mav De 
neglected, as e 2 is much smaller than kx 2 , so that 

u = — 7 -„ . - 9 co§ cot (33) 

faf CO 2 

The variation in the angular velocity has thus the same 
frequency as that of the rotation itself, but is very small in 
magnitude, as will be seen from the following figures. For 
a turbo-generator rotor balanced to about 1 oz. at radius 
A'i per ton weight of rotor, e\k x will be about 3 x 10 ~ 5 , and 
for a machine to run at 3000 R.P.M., £ x will be about 
50 cm., and co is, say, 2-7TX 50, 

3xl0~ 5 981 

U= ^r — x r—s — ^ — ^ cos cot 

oO 477-- x 50 x 50 

= 6xl0" 9 coso)£, 

that is the vibration is very small, the amplitude being only 
6 X 10" 9 of one radian. 

It should be noted that u is proportional to e and to g ; 
this variation therefore arises through the action of gravity 
on the rotor when not perfectly balanced, and the variation 
will be absent if the balance is perfect. 

Substituting the 'value of a in (28) and (29) in order to 
find the effect o£ the irregularity on the displacement : — 

e 2 
& + c 2 x = co 2 e cos cot + 7 2 g sin cot cos cot, 

e 2 
y -f c{ 2 1/ = co 2 e sin cot— — q cos 2 cot — g, 



136 Mr. C. Hodgers on the Vibration 

or 

1 e 2 
x + c?x — cw 2 e cos oot-i --=— g sin 2g)£, 

V + Ci 2 y = a> 2 e sin cot- - -^ g cos 2o>*-^ 1 + -^j . 

The solution is (for the forced vibration) 

co 2 e cos cot 1 e 2 1 . _ , rt JN 

I= l?^ + 2V^i' SmH (34) 

&> 2 £ sin &>£ 1 e 2 1 _ g /^ 1 e 2 \ , „ N 

The irregularity has thus two effects on the main whirl : 
firstly, the static deflexion is increased by a small amount, 
and secondly, there is superimposed on the main whirl a 
ripple o£ double frequency, which rises to a maximum at 
half the critical speed. But the effect is very small, and may 
not be noticeable ; in any case, as the double frequency effect 
depends on e it cannot appear when the machine is well 
balanced. 

There is, however, the possibility that a rotor which is not 
perfectly balanced may show vibration at half the critical 
speed due to the action of gravity, although gravity would 
produce no such effect at the full critical speed. A vertical 
spindle rotor is not of course subject to the action of gravity 
in this sense, and if it vibrated specially at half the critical 
speed, the cause must be sought elsewhere. 

13. We shall now consider some further possible causes of 
subsidiary critical speeds or speeds where marked vibration 
may appear other than the normal calculated critical speed. 

14. An important case is that of a rotor slotted for a 2-pole 
winding or with a shaft in which a key-way is cut, where 
the rigidity of the rotor is greater in one direction than in a 
direction 90° away, so that if the shaft is rotating, the stiff- 
ness in the direction of any one of the axes is not a constant er 
but a 4- e cos 2cot, where e is small in comparison with cr. We 
shall assume that the rotor is perfectly balanced (e = 0) and 
to simplify the examination shall first consider the vertical 
motion only. The equation is : 

M.y + cry=— M^— yecos2a>t. 

The first approximation is : 

y=— — = -#i- 



id Critical Speeds of Rotors. 



137 



Inserting this on the right-hand side of the equation and 
sol vino- we get 

y=— ' il-f - , M „gos 2 cot y 
,y a [_ cr — 4M&) 2 J 



= -^'i 2 { 



1 + 



€/'M 



cos 2w/ 



(36) 



Cl --4:C0 2 J' ' 

There is thus a double frequency vibration about the 
statically flexed position, which has a maximum when 
(0= 2 c u that is, at half the critical speed. 

It is evident that such a motion must have a tendency to 
arise if a rotor is unsymmetrical as regards its rigidity, for 
in such a case when the shaft rotates the deflexion will be a 
maximum or minimum twice every revolution, and if the 
frequency of the consequent up and down motion is equal to 
the critical speed there will be resonance ; this will be the 
case whether the rotor is perfectly balanced or not. 

It is thus possible for a perfectly balanced rotor which 
would be quite steady at the critical speed to show marked 
vibration at half that speed. If the normal running speed is 
above the critical the forces called into play at half the 
critical speed will be very small and may give no appreciable 
effect, but if the running speed is in the neighbourhood of 
half the critical speed vibration might arise. 

Fig. 4. 



u 



,--Ti£t 



15. It is worth while to examine the motion a little more 
fully as there will evidently be some vibration in the hori- 
zontal plane also. Let C (fig. 4) be the position of the 
centre line of the shaft and OA, OB two axes at right 
angles rotating about with the same angular velocity co as 
rotor. Let the co-ordinates of C be a and b with respect to 
OA and OB and u and v the corresponding velocities along 
those axes. 



138 Mr. C. Rodgers on the Vibration 

Then u =« — &©, 

and the accelerations are 

along A : u — vo) = a — 2/;<y — aw 2 , 

along OB : v + uco = b + 2aco — bay 2 . 

If the force required to produce unit deflexion in the shaft 
is <r + e along OA and a — e along OB, and we resolve along 
OA. and OB, the equations are : 

M(« — 2bco — aco 2 ) 4- (<r + e)a = — M<? sin g>£, 

M("i + 2aco — bco 2 ) + (o— e)6= — 1% cos cot ; 
that is, 

{M(D 2 -ft) 2 ) + (o- + e)}a-2Mr«)D/>=-M^sinft)^ 

{M(B 2 - co 2 ) + (a -6)}b + 2McoI)a=-Mc/ cos cot, 
giving 

[{M(D 2 -a) 2 ) + cr} 2 -6 2 + 4MVD 2 ] a 

= — M.g(a — e— 4&> 2 ) sin o>£, 
[{M(D 2 -o) 2 ) + cr} 2 -€ 2 + 4M 2 a) 2 D 2 ] 6 

== — M#(y + e— 4« 2 ) cos cot, 
The solution is (neglecting e 2 in comparison with cr) 

This gives the position with respect to the rotating axes ; 
the position with respect to the fixed axes is 



that is 



x = a cos cot — b sin cot, 
y = b cos cot -{-a sin cot, 



This result is the same for y as obtained in (34) by the 



and Critical Speeds oj Rotoi-s. 139 

method of approximations ; it shows that there is a similar 
motion of equal magnitude and 90° out of phase along the 
horizontal axis, so that the motion is a circular whirl of 
double frequency, which rises to a maximum at half the 
critical speed. 

16. When the rotor is out of balance the equation for the 
vertical motion is 

Mi/-f cn/ = M&> 2 £sin cot — Mg— ye cos 2cot. 

The first approximation is given by (25) and inserting this 
on the right-hand side of the above equation we get : 

1YT 
M y + crfj = Mmh sin cot — Mg + — e cos 2cot 

. e Mco 2 e , . . ' , / qo n 

+ 5" Tv/r «{sin m* --sin .•5ft>£|. \^) 
! (7 — Ivico 

The first three terms on the right correspond to the main 
whirl and the double frequency whirl already dealt with. 
The last term on the right will give in the solution a triple 
frequency vibration, viz. : 

1 Mor<? e . 

_ n M 2 nil/I 2 sin 6wt i 

I ff-Mr a — 9 Mar 

which has a maximum value at ^ the critical speed. This 
vibration is, however, proportional to e, the out-of-balance 
force, and cannot arise in a perfectly balanced machine. 
The remarks made as to the limited conditions under which 
the double frequency vibration might arise apply with even 
greater force to the triple frequencv vibration as the damping- 
effect of friction will be correspondingly greater. 

17. Another case of interest is that in which covers or 
sleeves are mounted on the rotor, or the rotor has slots in 
the periphery for an exciting winding, closed by pressed- 
in keys ; the closeness of these force tits will vary with the 
deflexion, and the deflexion of the shaft may therefore be 
not quite proportional to the force applied, i. e., the force to 
produce a deflexion % will not be o\r, but say a(x + ex 3 ), 
where e is small in comparison with unity. (The expression 
for the force must contain odd powers of x only as the 
rigidity is symmetrical, that is, the same numerically for the 
same numerical value of x whether x is positive or nega- 
tive ; if even powers were included this could not be the 
case as an even power of x is always positive even if ,c itself 
is negative.) 



140 Mr. C. Rodgers on the Vibration 

The equations then are, putting the small quantities on 
the right-hand side : 

x + c 2 x = (o' 2 e cos cot — c^ex?, 

y ~t" G iy — ( ° 2e *> m Mi —9 ~~~ c i 2 ^/- 

Neglecting the small quantities, the forced vibration is as 
before given by 

we 
X=-^-—: 2 COS(Dt,. ..... (40) 

9 

co e 
y=J*t^p ainht-g/d*. . . . . (41) 

Inserting these values on the right-hand side o£ the 
original equation, we get after some reduction : — 

x + c 2 x = {co 2 e - %<'i 2 ep 3 } cos cot — ^cfep 3 cos Scot, 

ij + Cl 2 y= -g + cfey^pP+yJ) + {co 2 e + c l 2 ep{'2i/ 2 t fp 2 ) } sin cot 

— Ci 2 €p 2 g cos 2cot - r jc^ep 3 sin Scot, 

where p= — J = 

c/ — OT 

and i /o =gjc l 2 . 

Solving these equations we get : 

* = /> ll-jc^e ^^ J cos tot-\c v 2 €-^_ — 2 cos 3u*, 

Ci 2 € 

2 • p 2 yo cos % w t 



C 2 -± 

* c i 2 e 

+ i • ~9 — rr~i P z sin 'dot. 
C] 2 - yco 2 r 

Examining these terms in turn we find that the centre of 
motion is now at the point 

#=0, 

instead of the point # = 0, y=- - y . This indicates that the 
centre of motion rises, i. e., the shaft straightens out slightly, 
as the vibration increases. 



and Critical Speeds of Rotors. 141 

The main vibration, represented by cos cot and sin cot, 
shows a slight change in amplitude, but as before the 
maximum occurs at the critical speed. 

The term in cos 2cot indicates there is a double fre- 
quency ripple in the vertical motion y (but not in the 

c 2 e 
horizontal component x) having an amplitude 2 2 /° 2 ffo? 

that is, ' l . ,.( ., .,) Vn- This rises to a maximum 

at half the first critical speed (when the amplitude changes 
sign) and again at the first critical speed (when the ampli- 
tude does not change sign). Noticeable vibration may thus 
occur at half the critical speed, but it will take place 
principally in the vertical plane. 

Both components show a triple frequency vibration ex- 
P'essed by cos Zcot and sin Scot, which reaches a maximum 
at one-third the critical speed, and the amplitude of the 
vibration changes sign at that point. This vibration also 
has a maximum value at the critical speed. 

Points of marked vibration due to lack of proportionality 
in the deflexion can thus only show themselves when the 
machine is out-of-balance, and if they become appreciable 
at all will only occur at half or one-third, etc., of the critical 
speed. If, however, these fractions of the critical speed 
correspond to low running speeds, the forces may be so 
small as not to produce any noticeable effect. 

18. It thus appears that subsidiary critical speeds are only 
to be expected at half or possibly one-third of the calculated 
first critical speed, and only then when the subsidiary critical 
speed is high enough to make the forces appreciable — for 
example, in the case of a turbo alternator when the speed 
indicated by the calculation approaches the running speed. 
All these effects should disappear with perfect balancing, 
excepting that due to lack of uniformity in the resistance 
of the shaft or rotor to bending in directions perpendicular 
to its axis, such as might arise through two-pole slotting of 
the rotor or through a key-way in the shaft. 

The forces tending to produce vibration are small, and the 
vibrations arise through a kind of resonance ; as there is a 
good deal of damping due to air friction and to the move- 
ment of the shaft in the bearing where the oil exercises a 
strong damping action, the vibrations may not arise at all. 
This question is gone into more fully in a later section. 

19. We have now to consider some cases where resonance 
may arise from causes outside the machine itself, and two 



142 Mr. 0. Rodgers on the Vibration 

classes may be noted, firstly where there is an irregularity 
in the torque applied to the shaft, and secondly where there 
is resonance with masses outside the machine. 

20. Irregularities in the torque driving the machine may 
arise, for example, through variation in the steam admission 
or through a fluctuating electrical load. 

The result of fluctuation in the torque will be a corre- 
sponding fluctuation in the angular velocity of rotation so 
that the angular position, instead of being cot, will be 
cot-\- € sin pt, where e is a small angle and p is an angular 
velocity corresponding to the frequency of the disturbance. 
Then cos {e sin pt) =1 and sin (e sin pt) = e sin pt. The equa- 
tions then become, taking the small quantities on to the 
right-hand side : 

x -j- c 2 x = co 2 e (cos cot — e sin pt sin cot) , 

jj -f c-iy = (o 2 e (sin cot 4- e sin pt cos cot) — g ; 
that is, 



x + c 2 x = &) 2 <?{cos cot + e/2 (cos o> +■ p . t — cos co — p . t) }, . (42) 
y -\- c^ 2 y — co 2 e{sin cot + e/'2{>in co-rp . t-\-s\\\co — p . i) }. . (43) 

The main vibration is the same as before, but there are 
two small vibrations superimposed ; the one has a frequency 
corresponding to co-\-p and a maximum when <w = 6 , 1 — p, the 
other a frequency corresponding to co— p and a maximum 
when a>=c 1 + p. 

This shows that the vibration may have a maximum at 
speeds corresponding to the sum of and to the difference 
between the critical speed and the speed corresponding to 
the frequency of the disturbing fluctuation. So that if dis- 
turbed running show itself at such a speed that it cannot 
be otherwise explained, a cause may be sought for in this 
direction. 

21. The other variety of resonance mentioned is that 
where, for example, the foundations are not sufficiently 
rigid and the machine as a whole is vibrating so that there 
is resonance between the rotor on its shaft and the machine 
on its foundations. A similar case would be that of a machine 
rotating in or near a building which itself shows marked 
vibration, possibly in certain parts only, corresponding to 
the vibration of the machine. Both these cases are similar 
in principle and may be illustrated by supposing the whole 
machine to be mounted on foundations having some elas- 
ticity. If then M x is the mass of the rotor and M 2 the 



and Critical Speeds of Motors. 143 

effective mass o£ the machine and that part of the founda- 
tions which moves with it, and the forces required to give 
unity deflexion are in the two cases cr, and a 2 respectively, 
the equations of motion are as follows : 

Mii/! + o-, (y, - y 2 ) == M 1& rV sin pt, 

M 2 i/ 2 + 0-22/2 - o"i (yi ~ Vi) = 0. 
giving for the amplitude of the forced vibration 

• • (44) 



where C\ 2 = 



a 



M,' 





u> 2 e(cx 


2 m + 


2 
'2 ~ 


CO 2 ) 


CO 4 - 


■ (o 2 {Ci 2 m-\-c 2 


+ «I 


2 J + CiV 


9 


°2 

~M 2 ' 


in = 


M 2 


• 



Points of marked vibration may thus occur at either of 
two frequencies given by putting the denominator = ; 

M 

these frequencies will therefore depend on the ratio ~ as 

well as on C\ and c 2 , and may thus have almost any values. 

For example, if c 2 = i\ and ?n = 0'2, i.e., the mass of the 
machine and foundations is five times the mass of the 
rotor — 

fi> s g(l-2c 1 2 ~o> 2 ) 
• y ~a> 4 -co 2 (• 2 2 x2•2 + r I 4, 

and a maximum occurs when 

ta=c 1 xl'25 or c 1 x 0*80, 

that is, at speeds 25 per cent, above and 20 per cent, below 
the calculated critical speed. 

If M 2 , the mass of the machine and foundations, is very 
large in comparison with M 1? the mass of the rotor, the 
denominator is very nearly equal to (co —c{) X (a> — c 2 ), which 
shows that in such a case the two speeds where marked 
vibration may occur nearly correspond to the natural fre- 
quency of the rotor and of the machine and foundations 
respectively. But as the numerator is also small the vibra- 
tion might not appear if considerable friction is present. 

It: vibration should occur when ft) = c ] /v / 2, which is, as 
mentioned above, sometimes thought to be a critical speed, 
this might indicate that there was resonance with the founda- 
tions or some structure outside the machine, in which case, 



144 Mr. C. Rodgers on the Vibration 

putting co 2 = c 1 2 /2, 

c 1 2 -2 W{l + m) + r 2 2 } -f4c 2 2 = 0, 

that is c 2 = \ ' — g — c„ 

so that if, for example, M 2 were large in comparison with 
Mi and m is therefore small, 

c 2 = c 1 /V2 = c l xO-707, 

but if, as in the former example, M 2 = 5M l5 

c 2 = CjL x 0-836. 

B. Transverse Vibrations with Friction. 

1. The frictional resistance opposing the motion of the 
rotor may be considered to consist of two parts. The first 
part opposes the rotation of the rotor about the centre 
line, C, of the shaft, and this is counteracted by the torque 
supplied by the turbine, or, if the turbine is cut off from the 
steam supply, it tends to bring the set to rest ; it has no 
retarding effect on the whirling. The second part opposes 
the whirling only, and it is with this that we have to deal. 

2. It is not known how the frictional resistances opposing 
the whirling vary with the speed, but it seems likely that 
they vary with the square of the speed at least. We shall, 
however, first consider the case where the resistance is 
assumed proportional to the first power of the speed, as the 
motion is then simpler to work put, and there is an inter- 
esting electrical analogy, which enables the motion to be 
more readily followed. 

The resistance to whirling is in opposition to the path of 

ds 
the rotor centre, so that if -r- is the speed of the centre in 

any direction, the frictional resistance is M//,f , ) , and the 

(ds\ n dx 
7 - ) t- and 

f(xs\'" av 
M-fjulj.) -/-, that is, Mfzs^x and M/^" 1 ?/, where fi is a 

constant. 

3. In the particular case we are about to consider, n — 1, 
and the components are therefore M/xx and Mfiy. 




and Critical Speeds of Rotors. 145 

The equations of motion are therefore 

Mai -I- M/Xci; + ax = Ma) 2 e cos tot, 

M.y + Mfiy + ay = Mco 2 e sin atf — M#. 

Electrical engineers will notice the similarity between 
these equations and 

Jj'q -f ~Rq + p- g = E sin o)£ + E 0j 

which holds for a circuit comprising an inductance L, a 
resistance JEfc, a capacity K, an alternating E.M.F. of 
maximum value E, and a steady E.M.F. E , q being the 
charge in the condenser at any time. Thus, mass is equiva- 
lent to inductance, capacity to deflexion per unit force, and 
applied E.M.F. to applied mechanical force. 

The solutions of these equations are, as is well-known : — 

q = ~$e~ t > T sin (pt — <f>) 

E • / , ■ i E \ 

H r /-. it- t — ro — TV^r sl n ( cot — tan zmy 7 — 

V{(l/Ka)-La)) 2 +li 2 } V 1/Kft) — La>/ 

+ KE , (45) 

^ = Ne _i/T sin (pt — <f>) 

Mco 2 e . / ' , _, Maw \ 

+ ^{(,-M.f+MV^} sm r " tan " ^Mo?j 

-M<?/<r, (46) 

that is, 

r/ = Xe~ f/T sin pt — <j> 

+ /7-5 — -072 , » 2 sin ( wt -tan" 1 2 A6> 2 ) — ^/q 2 , 

, • • • ( 47 ) 

where 2L 2M 



^ = Vl/LK-1/T 2 or W/M-l/T 2 , 

There is therefore in both cases a free vibration having a 
frequency slightly less than the natural frequency of the 
system, but independent of the frequency of the applied 
E.M.F. or of the speed of the rotor. This vibration is 
damped out by friction. 

Phil. Mag. S. 6. Vol. 44. No. 259. July 1922, L 



146 Mr. C. Rodgers on the Vibration 



© 



The forced vibration is permanent and has a value which 

is proportional to the applied E.M.F. or out-of -balance force. 

The vibration is a maximum when co = c u that is the critical 

speed is the same as the stationary critical, as in the case 

where friction is ignored ; the amplitude of the vibration at 

.Ee.. 
the critical speed is p or — , that is, is equal to the applied 

E.M.F. or out-of-balance force and inversely proportional 
to the resistance. 

The frequency of the forced vibration is the same as the 
frequency of the E.M.F. or of the rotation, and the charge 
or displacement lags behind the E.M.F. by an amount 
depending on the frequency or speed and on the capacity 
and inductance or elasticity of the shaft and mass of the 
rotor. 

The lag is zero when the frequency is low, but increases 
to 90° at the critical speed, which, as will be seen, is that 
corresponding to the natural frequency of the system, while 
at very high speeds the lag increases to 180°, in other words 
the force is in opposition to the displacement. The change 
is similar to that occurring when there is no friction except 
that in the present case the change is gradual instead of 
taking place suddenly at the critical speed. 

It will be noted that in both cases the vibration takes 
place about the statically deflected position as a centre. 
It is evident in both cases that a large static deflexion 
would increase the tendency to break down, in the one case 
by puncture or flashing over of the condenser, and in the 
other by fracture of the shaft. 

4. If the resistance to whirling is proportional to the 
square of the speed, that is n = 2, the equations are : 

x + fi'sx 4- CiX — co 2 e cos cot, 
y + fisy + c 2 y = co' 2 e sin cot — g. 

The free vibration (i. e. the vibration when g = or the 
rotor is perfectly balanced) cannot be expressed in simple 
terms, but as it will be damped out, as before, it is not of 
interest. 

The forced vibration (i. e. the vibration due to the rotor 
being out-of-balance) is given by 

.£ = R cos (&>£ — <£), (48) 

y = R sin (cot - (/>) — g/ Cl 2 , .... (49) 



and Critical Speeds of Rotors. 147 

where R is the radius of whirl of the value : 

co e 

and tan<f> = \ 9 (51) 

r Ci— co z v y 

The lag of the displacement behind the force is in this case 
proportional to the actual deflexion, and in this respect differs 
from the result obtained in (47). This is of interest as it 
shows that since the radius of whirling is for a given speed 
dependent on the out-of-balance, the phase lag will be smaller 
the more perfectly the machine is balanced ; in the former 
case where the friction varied as the first power of the speed, 
the lag was independent of the amount of out-of-balance. 

The maximum deflexion occurs, as before, when &> = c l5 that 
is, when the speed is equal to the stationary critical speed. 

5. It is impossible to draw any conclusion from these 
formulae as to the real angular advance corresponding to a 
given speed, as it is not known how the frictional resistance 
varies with the speed. We can, however, say that if the 
machine is rotated first in one direction and then in 
the other, the position corresponding to the out-of-balance 
will be mid-way between the points of maximum deflexion. 
When balancing a machine in the running condition it is 
usual to hold a pencil or chalk against the shaft so that a 
mark is made on the shaft at a point corresponding to the 
maximum deflexion. If there were no friction and the speed 
were not the critical speed, this mark would be in phase with 
the heavy side of the rotor below the critical speed, and 180° 
oat of phase with it if above the critical speed. But it will 
be seen from (51) above that the actual position of the mark 
depends both on the amount of friction and on the amount 
of out-of-balance. At the critical speed the heavy side of 
the rotor should be 90° out of phase with the mark on the 
shaft, but the actual position will be uncertain, as the angle 
varies rapidly with departure from the critical speed, and it 
is not usually possible to judge exactly when the machine is 
running at the critical speed. 

6. There is another reason why the position of the mark 
on the shaft is somewhat uncertain. Referring to fig. 3, if 
we ignore all other vibrations than that corresponding to the 
variation in <j), we get by taking moments about G : 

MkM+crre sin 4> = (52) 

L2 



148 Mr. C. Rodgers on -the Vibration 

If <j> is a small angle this becomes : 

M/q 2 <£ + am/)=0, 

the solution of which indicates a periodic motion having a 
time o£ vibration of 



T = 2tt^ 



Mk, 2 



= 2 *V§e < 53 > 

In an actual machine for 3000 R.P.M. we shall have 
figures of the order of: — 7^ = 50 cm., k A \e — 3 x 10 4 , 
c 1 = 27rx30, say, while r may be of the order of 1 mm., 
so that 

T __ 2it /50 x 3 x 10 



2vrx30 V 0-1 

= l'4Xl0 2 sec. or about 2 mins. 

As the time of vibration is very long in comparison with 
that of the other vibrations occurring, it will be almost 
unaffected by the latter, and the assumption that the other 
vibrations can be ignored, which was made in deducing 
(52), is therefore justified. 

For larger values of e corresponding to less perfect balance 
and for deflexions of greater magnitude, T will be corre- 
spondingly less, and will be greater the more perfect the 
balance. If friction is ignored, r becomes infinitely great 
at the critical speed and T becomes zero, and although this 
can never be the case in practice, it is clear that T may have 
a value of two mins. or more down to something considerably 
smaller. 

In other words, if the rotor is disturbed from its position 
of equilibrium by any chance external cause, it may take a 
considerable time to settle down, and during that period the 
position of the mark on the shaft will vary considerably 
from its normal position. 

7. At the critical speed the lag is 90°, and the vibration 
is also a maximum, but the sharpness of this maximum will, 
as indicated above, depend on the frictional resistance to 
whirling. In addition to this it will also be influenced by 
the condition and design of the bearings, as the oil in tha 
bearings exercises a considerable damping influence and aL-o 
introduces a further complication as follows : 

When the speed is low and the vibration therefore small 



and Critical Speeds o) Rotors. 149 

in magnitude, the film of oil in the bearings will allow the 
shaft a certain amount of play ; this will increase the effec- 
tive length of the shaft and lower the critical speed; with 
increasing speed the vibration will therefore start up fairly 
smartly. As, however, the speed increases and the vibration 
becomes greater, the shaft may bed hard up against the 
bearing bush, and increased deflexion will decrease the effec- 
tive length of the shaft, and so raise the critical speed. As 
tbe speed is further increased a similar state of things is 
gone through, so that at a certain point the vibration will 
die down more quickly than if there had been no film of oil. 
The effect of the oil in the bearings is thus to give an 
added amount of friction to whirling, and at the same time 
flatten the maximum peak of the vibration, that is, the 
vibration will start up and cease fairly smartly and remain 
more or less constant throughout a fair range of speed. 
If, however, the film of oil is sufficiently thick or the 
balance sufficiently good, the vibration may not show itself 
at all, although it might do so with the same out-of-balance, 
if the film of oil were thinner. 

Section III. — Oscillatory Vibrations — Second 
Critical Speed. 

1. Oscillatory vibration may arise in two ways, either 
through lack of balance or through vibration transferred 
from the transverse motion. 

2. The lack of balance referred to is of the skew type, that 
is, is equivalent to a pair of weights at opposite ends of the 
machine, and 180° apart, giving an out-of-balance couple 
when the machine rotates ; such an out-of-balance will not 
show itself when the machine is being statically balanced on 
knife edges, and can only be corrected through observations 
when the machine is running. 

3. Vibration can be transferred from the transverse motion 
only when the machine is unsymmetrical in the sense that a 
force applied to the centre of gravity at right angles to the 
shaft gives a displacement which is not parallel to the centre 
line, that is, in those cases where, on the static deflexion 
diagram, the shaft in the deflected position is not parallel to 
the centre line of the bearings. 

4. The form of out-of-balance mentioned will produce a 
couple rotating with the machine, that is, a couple alter- 
nating with the frequency corresponding to the running 
speed. Vibration transferred from the transverse motion 
ma}-, however, be of the frequency corresponding to the 



150 Mr. C. Rodgers on the Vibration 

speed of the machine, but may also be of double frequency 
arising through any of the causes we have discussed. 
Further, when the normal frequency oscillation has estab- 
lished itself, a double frequency oscillation may start up 
owing to bipolar asymmetry or some of the other causes 
mentioned in connexion with the transverse vibration. It 
is therefore necessary to consider in the oscillatory motion 
forces both of the actual frequency of rotation and of 
double frequency. 

5. We have found that the transverse motion can, with 
sufficient accuracy, be considered the same as for a perfectly 
balanced machine with an out-of -balance force attached to 
it. In the same way we shall treat the oscillatory motion as 
being due to an out-of-balance couple of the frequency corre- 
sponding to that of rotation or a multiple of that frequency 
acting on an otherwise perfectly balanced machine. 

6. In the diagram fig. 5 let G be the centre of gravity of 
the rotor and GL the direction of the centre line of the rotor 




twisted from its normal position by an angle f = L(}Z, where 
GZ is the direction of the centre line when not vibrating. 

The direction cosines of the centre line GL with the axes 
GX, GY, and GZ are respectively f , 77, and J ; if L is a 
point at unit distance along the shaft from the centre G, 
then J, 7), and f are also the co-ordinates of the point L, as 
shown on the diagram. 

If the moments of inertia of the rotor about the shaft 
centre line and about the line at right angles to it, through 



and Critical Speeds of Rotors. 151 

the centre of gravity, are A and B respectively, the angular 
momenta are *: 

about GX, ^B^-SHAwf, 

about GY, ft 2 =B(?f-ff) + A©?7, 

about GZ, A 3 = B(^-^) + Ao)f. 

In the actual case the angle yjr is very small, so that we 
can put sin-v/r = ^ cosi/r=l, and f=l, f=0j also the 
products f 77 and 77^ are both negligibly small. 

We thus get : 

7*i =— B?7 + A&)f, 

7l 2 =:B| + Aft)>7, 

7i 3 = Ao). 

* Another and perhaps more legitimate way of deducing these 
equations is as follows : — 

If a lt wo, and o) 3 are the instantaneous angular velocities about moving 
axes 3rX', GY', and GL fixed in the rotor and moving with it : consider 
the iistant when GX' is perpendicular to GL and GZ (cf. fig. 5), and 
let )e the angle between the planes LGZ and YGZ. 

Th«nw l =— -^ w 2 = 0sin^, and w, = w. 

Th( angular momenta about GX', GY', and GZ are Bo>i, Bw 2 , and 
Aw. The anuular momenta about the fixed axes GX, GY, and 
GZ aie : 

Ai = B{ w l cos 0+w 2 cos \p sin 0f4 Aw 3 shuf/sin 9, 

h 2 = B { — o>i. sin 9 4- w 2 cos \p cos 9 \ -j- Aw 3 sin ^ cos 0, 

7i n — B{ — w 2 sin^}+Aw 3 cos^ ; 

that i; 

hi= B{— ^ cos 9-\-9 sin ip cos 4> sin 0} + Aw 3 sin;// sin 9, 

h 2 = B{ -4/ sin 0+6/sin^cos4/cos0} + Aw 3 sin^cos0, 

h = — B0 sin 2 iff+Awj cos 1//. 

Al.o 

S = sin^sin0 and £=ij/cosi//sin 0+0 sin ^cos 9, 

J7 = sin'4/cos0 and r/ = ^ cost// cos 0—0 sin ^ sin 9, 

£ = cosip. and £== — i£sin^; 

so thit 

tlK—Ki = —yf cos 0+0 sin cos^sin0. 

Zi-ZZ— i\j sin 04-6* sin ^ cos $ cos 0. 

% n — ■,)'%= — O&m 2 ^. 

ty substituting these values in the equations for A t , A 2 , and A 3 , we 
obain the relations given above. 



152 



Mr. 0. Rod o'er s on the Vibration 



(54) 
(55) 
(56) 



The force o£ restitution of the shaft is : 

about GX = ^, 

about G-Y= -—«£", 

about GZ=0. 

If the out-of-balanee couple is ra 2 , the moment of the 
couple is : 

about GX = — ra 2 sin pat, 

about GtY = to) 2 cos pat, 

about GrZ —nil, 

where p = l or 2. 

The equations of motion are thus : 

— B rj — fcr) + Ao>f = — ra 2 sin pat, 

Bf+/cf + Aft)^ = TCO 2 COS Jt?&)£, . . 

Aai=0; 

or putting D for oj/d£, 

(D 2 + c 2 2 )rj — maD^= rco 2 /B . sin pat, 
(D 2 -}-e 2 2 )f -f mo)D^ = T&) 2 /B .cospw^ 
Ai = 0, 

where c 2 2 =^ and m=~-. 

The last equation A&> = gives -co = const., that is, trBre is 
no fluctuation in the angular velocity of rotation. 
The other equations give 

{(D 2 + c 2 ) 2 + m 2 a> 2 D 2 } f = ra 2 /B {{c 2 2 -p 2 a 2 )-mpa 2 } co\pat, 

and a similar equation for f in terms of sin pa)£. , 

7. The free vibration is not of importance, as it wil be 

damped out as before, but it is of interest to examira its 

value as illustrating the effect of gyrostatic action oil the 

motion. 

The free vibration is of the form 

x = Ni sin (qt — (f>i) ; y — N 2 cos {qt — cf) 2 ) , 

where N 1? <f>i, N 2 , cf> 2 are constants and 



+ q — ±{ma+ \/m 2 a 2 -\- 4c 2 2 }. 



'7) 



and Critical Speeds of Rotors. 153 

There are thus two natural frequencies of whirling, 
depending on the direction in which whirling takes place. 
For example, if the machine is running at what we shall 

presently see is the critical speed, namely, mo) = c 2 — \/ K , 

there are two possible frequencies of whirling for the free 
vibration, viz. : g=l*618 C 2} or 0*618 C 2 . 
8. The forced vibration is given by 

* ™V B 



» (J ,B- A) -Btf. . «****'•' * • (57) 



poa 

with a similar equation for r; in terms of sinpcot. 
The amplitude of the vibration is a maximum when 

po> 2 (pB-A)-Bc 2 2 = 0, 
or 

E) 2 fe 2 = / V A^ = I 1 M • < ' ( 58 ) 

' 2 p(pB — A) p(p-m)' 

9. This enables the second critical speed to be calculated 
without difficulty, and fig. 6 gives the necessary curves for 
reading off the proportional values directly. 

The ordinates give the values of co/c 2 and the abscissae the 
values of the ratio A/B. The method of using the curves is 
as follows : — 

1. Work out the radius of gyration ki about the shaft centre 

line. 

2. Work out the radius of gyration k 2 about a line perpen- 

dicular to this through the centre of gravity. 

3. Work out the ratio of A/B or k 1 /k 2 ; for turbo-generator 

rotors its value is usually between *2 and *4 and for 
flywheels up to 2'0. (This gives the working point on 
the horizontal axis of the curve.) 

4. Work out the *' first " critical speed in the usual way and 

multiply by ljk 2 , so as to obtain the stationary "second " 
critical (where Z = half the distance between bearing 
centres). 
•5. To obtain the second critical speeds read off from the 
curves the figures given on the vertical axis and 
multiply by the stationary critical found from (4). 



154 



Mr. C. Rodgers on the Vibration 



10. The couple causing oscillatory motion will usually be 
of a rotating type having components represented by the 



Fisr. 6. 



2-5 








rax 


.': 1". ■'.:!. • ■ 1 : > ', > ' i 


.. ...,_,....!. i : ,.j_.I .;...,... |._i..:. . . ,. ..._.^.L_ 


1 

: 

• 

• 


i ' i ' 


-_.Li_!:".U_.i 1 .: -.-.J : . : i/-i It i: - . 




r r 




■ 1 1 




!i_] 


; J j :' 


. | i. " 1 . ' l" ; l~~ : . !' "} .-: }-:[: ■". 




_ : 






** 


















_______ .. 


...i . . ..;... . ,. . /..[. ...]„L_ 




V't-r- 


r/ r " : i t "j 


•T-™p j ■ ' - 






S.-_o^_7-.7'\.7 Sr-jx«£^rT:~- ' 7 ;'7"" '" 






j , ! : ... ;.. . j ,'. 1 ... . 


hj< — l^te - 




.: ; . •: / ■ ■ 


h(A—~ i -j^ttfjf- 




--!■ ■ ; : i J, - : - ■ 


*°|n'-'2"T''~ 






• J • pr ; _ 


...... . _| 




y.jL.^...- . 








. /' ; "". i ' " ! 


..:_... ;.. _• .. . .. i . : ,• 


i'2 


; ! ; 


TP'l • : 




H 

ho 


_ [ ..'.'• ^ 






PPj- 












'9 


7.7- )_T~ 






*? 








1 


.... ,. . : J . . j 




..s.j 


■i 

7|7„ 

;„j-... 


T77] — 


... . :. . . . ... • . .|_al_J_!. 






"TT ~"~~ |~r~~ * •-' ; . 






tijttc 






'•' 


• ■ ! hi ! : ! ; 




! 6 
l 1 


! : 
Li L ; 


-:— | ■■--•- ;• ']'— ? ■ ■+•• ; ' f. - 




jftj- 


1 7 4 B :. /' I 














>:. .,.:/ T"_> ""V""^. -5 •_ 7 V""-?, /•X"'^ A3 ~* >.> /■_ ■ ^ ^.'; A ? -F 


1 



right-hand side of equations f54) and (55). It is, however, 
conceivable that one of these components may be absent if 
the disturbing couple is due to some external cause. Thus 
if there is a couple round the x axis only, equations (54) and 



(55) become 



and Critical Speeds of Rotors. 155 

— B?; — kv + Aw£= — 7(o 2 ±h\pcot, . . (59) 
Bf+tff + Ao^O, (60) 



giving the forced vibration 



9 <> <> 2 

V ~TT (c 2 2 -jpV) 2 -mV> 4 



input. . . (61) 



sin 



The oscillatory motion is thus a whirl the nature of which 
depends on the speed. The whirl is a maximum when the 
denominator is zero, that is, at two speeds, one on each side 
o£ the stationary critical, and given by 

a ^ = ^— (63) 

There is thus a possible further second critical speed, 
corresponding to the + sign, lower than that already found 
corresponding to the — sign. 

For the ordinary rotating couple the direction of whirling 
is, of course, always in the direction of rotation, whether the 
speed is above or below the critical, and this is indicated by 
the fact that, as will be seen from equations (58), the sign of 
the amplitude in both planes changes, showing also that the 
phase of the motion has changed by 180°. But for an 
alternating couple about the x axis only, as will be seen from 
(61) and (62), the motion in the horizontal plane changes 
sign at each of the critical speeds indicated by (63), while 
the motion in the vertical plane has a further change of sign 
when p 2 co 2 = c 2 2 . It will be seen, if these changes are fol- 
lowed out, that the whirling is in one direction below the 
stationary second critical and in the opposite direction above 
that speed, while at the stationary second critical the motion 
is in the horizontal plane only, that is, at right angles to the 
applied torque. 

11. The following example is given to illustrate the appli- 
cation of the above curves and formulse. 

A rotor consists of a solid cylinder 30 ins. diameter and 
60 ins. long, running in bearings, 107 ins. between centres. 
From the deflexion diagram, the deflexion at the centre of 
gravity is, say, *0087 in. or '0221 cm. 



156 Mr. P. Cormack on Harmonic Analysis of 

We then have : 
The first critical speed c x is -^L =2020 R.P.M. 

The ratio of the radii of gyration k 1 jk 2 is 6/19 = 0*316. 

Half the distance between bearing centres = 53*5 ins. 

The radius of gyration for the cross moment of inertia 
k 2 = 18-87 ins. 

The stationary second critical is, therefore, 

6' 2 = Cl x ^=- = 5730 R.P.M. 

From the curves, fig. 6, it will be seen that the main 
oscillation will occur at 

5720x1-21 = 6940 R.P.M. 

and that a double frequency vibration may possibly show 
itself at 

5730 x 0-515 = 3120 R.P.M. 

Thus a turbo-generator designed for a speed of 3000 R.P.M. 
and having the above mechanical constants might show 
marked vibration on the overspeed test. 






X. Harmonic Analysis of Motion transmitted by Hookers 
Joint. By P. Cormack, A.B.C.Sc.L, Engineering Dept., 
Royal College of Science for Ireland *. 

1. ^1|TITH the growth of high-speed machinery, the 
H determination of the accelerations of machine 
pieces becomes of increasing importance. These deter- 
minations are considerably simplified by expressing in the 
form of a Fourier Series the displacement of the piece 
under investigation. The value of this method in the 
analysis of the various phases of the motion of the mechanism 
of the direct-acting engine is well known. It is here 
proposed to investigate the coefficients of a Fourier Series 
for the angular displacement of the driven shaft of a Hookers 
Joint. The method being applicable to certain inversions of 
the slider crank chain, these are also included. The ease 
with which the coefficients can be determined, and calculations 
made from the resulting series, make the study of these 
mechanisms from this aspect one of considerable interest. 

* Communicated by the Author. 



Motion transmitted by Hooke's Joint, 



157 



2. In Hooke's Joint the point B moves in the great circle 
CBN and the point A in the great circle CAN (rig. 1). The 
arc AB is constant and equal to a quadrant of: the great 
circle. The point A receives its motion from an arm OA 




set at right angles to the driving shaft OX, while B transmits 
motion to the driven shaft OY. Let the angle between the 
shafts be y • this w r ill be the angle between the planes of CAN 
and CBN. In the spherical triangle ABC we have 

cos <• = cos a cos b + sin a sin b cos 7. 

Since c=7r/2, this becomes 

cos a cos Z> + sin a sin b cos y = 0. . . . (1) 

Plainly when B is at C, A will be at T; B will therefore 
move through the angle a while A moves through the anple 

&-7T/2. 

Writing <£ for a and 6 for b — tt/2, equation (1) becomes 
— cos (/> sin 6 + sin <f> cos cos y — 0. 

Put cos7 = (l — »)/(l + n), and we have 

(1 + n) cos </> sin 6 = (1 — n) sin cj> cos 6. 

.*. 11 (cos (/> sin 9 + sin (/> cos 6) = sin <£cos 6 — cos </> sin #. 
sin (<£ — 0) = n sin (</> + 6). 



158 Mr. P. Cormack on Harmonic Analysis of 

Multiplying both sides by 6^ -0 ) gives 

e 2K<p~e)_ 1==ne 2i<i>_ ne -2ie t 



... e 2i (t-^{l~ne 2i9 \ = l- 



ne 



-2i9 



Mt-0) = (l-ne- 2ie )l{l-ne™). 

IT , 7T 

- and — y 



Since (p — lies between -f- and — ^, and n is less 



than unity, we may write 

2i(cj>-6) = log (l-^- 2 *' )-log (l-ne 2i9 ) 

= - ne -2i9_±_ n 2 e -ii9_i n 2 e -QzO_ t < t 

+ ng 2 ^ + \n-e^° -f- •fn 3 *? 6 *' + . . . 

= n. 2i sin 20+ ^n 2 . 2z'sin4(9 + §n 3 . 2i sin 6(9 + . . . 

... <£-# = w sin 26 + \n 2 sin 4(9 + l?z 3 sin 60 + (2) 

It will be evident that (2) gives the displacement of the 
driven shaft relatively to the driving shaft. In practice 
the angle between shafts joined by a Hooke's coupling 

rarely exceeds 15°. Since cos a = r , we have 

J 1 + n 

n — (l--co3a)/(l + cosa) = tan 2 -. 

For a=15° we get w = '0173, so that we can without 
appreciable error neglect the terms containing the square 
and higher powers of n in (2) and put 

$-d = 7i sin 26 (3) 

For the above value of a, the maximum value of </> — 6 
given by (3) is "0173 radian or nearly one degree. 
From (2) we have 

</> = + n sin 26 + \n 2 sin 4(9 + |n 3 sin 66 + (4) 

dfy = /^\ 2 ( _ 4?zsin 26-8n 2 sin4(9-12?i 3 sin 6(9-...). 

• • • W 

In obtaining (5) and (6) we assume the series formed by 
the term-by-term derivative of the member on the right 
in (4) and (5) to be convergent and to converge to the 
differential coefficient of the member on the left. In 



Motion transmitted by Hookes Joint. 159 

obtaining (G) we take the case in which the speed of 
the driving shaft is uniform. 

From (5) the maximum speed of the driven shaft 
is readily seen to be co{l + 2rc/(l — ?*)} or w sec a ; the 
minimum speed is seen to be <w{l — 2n/(l-\-ii)} or « cos a, 
a) being the angular velocity of the driving shaft. 

When the angle between the shafts is not large, (G) may 
be written 

"(a)' w 

Thus the maximum angular acceleration of the driven 
shaft is very approximately 4?z&> 2 . If a = 15° and co = 
GO radians per sec, the maximum angular acceleration 
is almost 250 radians per sec. per second. 

Fig. 2. 



d *<t> • A 

t^- = — 4n sm 

dt 1 ' 





3. In the mechanism of the oscillating cylinder engine, 
and the quick return crank and slotted-lever mechanism 
(fig. 2), we have 

sin <f> _ sin ty _ sin (# + <£) 



or 



sin (p = -sin (6 + (f>) = n sin (# + </>), where ?i= -. 
c v 



g2ty(l_ ne *e) = l_ w < 



/6> 



,2/0 _ 



(l-ne- i9 )/(l-ne ie ), 



160 Harmonic Analysis of Motion 

t; 

2 



Since <fi lies between +— and — - , and n<l, 



2icf> = log (l_^-^)_log (1 -ne iQ ) 
= -ne- ie -±n 2 e- 2i9 - : k,nh- 2>i9 -... 

+ ne ie + We 2ie + ^nh™ +'... 
= n . 2z sin + in 2 . 2z sin 20 + Jn 3 . 2i sin 3(9 + . . . . 

<£ = ttsin0 + in 2 sin20+§n 3 sin30 + (8) 

^ = (?icos0 + n 2 cos20 + n 3 cos30 + ...)^. . • (9) 

^ = (_» sin 0~2n 2 sin 20- 3n 3 sin 30-...) fc 

. . ! (10) 
R* 3. 







4. In the Pin and Slot mechanism (fig. 3) we have 

"\|r = IT — — (£. - 

The angle <£ is given by (8), so that 

n ii 

^ - rjr-O-n sin 0- T sin 20-^- sin 3(9- ..., (11) 

<i± _ _^ (1 + n C os (9 + 7i 2 cos 2(9 + n 3 cos 30 + ...), (12) 
ttt etc 

^ = (~ V (n sin + 2n* sin 20 ■+ 3?* 3 sin 30+ ...). (13) 

It will generally be found that we need to consider but 
the first few terms of these series in making numerical 
calculations. 



L 161 j 



XL Short Electric Waves obtained by Valves. By E. W. 
B. Gill, M.A., B.Sc, Fellow of Merton College, Oxford, 
and J. H. Morrell, M.A., Magdalen College, Oxford *. 

1. f INHERE have recently been discovered methods for the 
i generation of continuous oscillations of short wave 
length (of the order of about a metre) by means of three 
electrode valves. In January 1920, Barkhausen and Kurz f 
found that with hard valves — i. e., valves at extremely low 
pressure, if the filament and the plate were approximately 
the same potential, or, indeed, if the plate were at a potential 
considerably lower than the filament, provided that the grid 
was kept at a high potential with regard to them, continuous 
oscillations could be maintained in a circuit of the Lecher 
Wire type connected to the grid and plate. The wave length 
depended primarily on the grid voltage, but also on the 
emission from the filament and on the plate voltage. 

Whiddington J had previously described another method 
of getting oscillations of lower frequencies using a soft valve, 
i. e.j a valve containing gas at low but appreciable pressure. 
He employed more usual circuits for a valve, in that the 
plate was at a high positive potential with regard to the 
filament and the grid at a few volts above the filament. In 
this case longer waves were emitted, and he noticed that if 
V was the grid potential and \ the wave-length emitted, 
then X 2 V was constant §. 

There appear to be other arrangements not hitherto re- 
corded which will also give these waves. With a hard valve 
and with the grid at a positive potential, oscillations can be. 
obtained if the Lecher Wire system is connected across the 
filament and grid.; the plate may be positive, negative, or at 
the same potential as the filament, or it may be insulated. 
Further, the third electrode — the plate — is unnecessary, for 
oscillations can be sustained by means of a valve consisting 
of a filament and an anode formed as a spiral of wire con- 
centric with the filament, when these two are connected to 
the Lecher wires. An intermediate arrangement has been 
worked successfully in which the wave-length of the diode 
connected as above is modified by a cylinder concentric with 



* Co mm uni cited by Prof. J. S. Townsend, F.R.S. 
+ Phifsikalischer ZeiUchriftj Jan. 1920. 
X AY biddington, ' Radio Review,' Nov. 1919. 

§ For a general account of these experiments see ' Radio Review, 
June 1920. 

Phil Mag. S. 6. Vol. 44. No. 259. July 1922. M 



162 Messrs. E. W. B. Gill and J. H. Morrell on Short 

the anode, but placed outside the valve and set at various 
potentials. The best conditions for these cases are still 
under investigation. 

2. Barkhausen and Kurz were apparently unable to give 
any explanation of the way in which the oscillations were 
sustained, while Whiddington assumed that the emission of 
ions from the filament was discontinuous and occurred in 
bursts. The authors, on the other hand, do not think that 
any special assumptions are necessary, and that the ordinary 
conditions for the maintenance of oscillations by continuous 
emission will account for all the facts they have observed, 
provided that the time taken by the electrons to pass between 
the electrodes is taken into consideration, as this time is of 
the same order as the period of the short waves. 

In the present paper only oscillations of the Barkhausen 
type are considered in detail, but the theory can be extended 
to cover all the types, and an account of some experiments 
on the last type (with a diode) will be published later. 

It is worth noting that certain writers give the impression 
that the seat of the oscillations is in the gas or in the 
electrons in the valve, and that the Lecher wires connected 
to the valve serve only to demonstrate their existence *. It 
appears from our experiments that the wires or conductors 
attached to the electrodes are a necessary part of the 
oscillatory system. Even with the Lecher wires removed, 
there will always be some circuit composed of the connecting 
wires to the batteries or even the valve leads up from the 
sockets, which will have natural periods of a suitable order 
for short wave oscillation. This fact seems to have been 
overlooked in some recent determinations of ionizing poten- 
tials, where large emissions from a heated filament were 
used as a source of electrons. Oscillations will take place 
even when the valves contain a small amount of gas, but in 
all the experiments described in this paper gas-free valves 
were used. 

3. It will probably be most convenient first to describe the 
experiments in detail, and then to set out the theory and 
apply it to the observed facts. 

Various valves were used, but mostly the Marconi M.T.5 
valves, which were very kindly given to us by the Marconi 
Company. These valves consist of a straight filament FF 
held in the centre of the valve by springy arms. The ad- 
vantage of the spring is that when the filament is heated and 

* Whiddington's theory is independent of there being any external 
tuned circuit. 



Electric Waves obtained by Valves. 



163 



expand?, the spring prevents sagging. Surrounding the 
filament is a cylindrical wire grid, GG, composed of thin 
wire of square mesh, each square having a side of about 
1*5 mm. The lead to the grid goes out at the bottom near 
the filament leads. A cylindrical plate, PP, surrounds the 
whole with its lead going out through the top of the bulb. 
These valves being used for transmitting purposes are very 
thoroughly "glowed out * and pumped to a very high vacuum. 
The filament emission is very high when heated with 6 volts 
direct, and for the low emissions that were generally used it 
was very constant. As the plate lead passes through the top 
of the bulb, instead of through the bottom and the sealed 



Fisr. 1. 




socket, very high insulation is obtained, and, if a strip of 
tinfoil connected to earth is placed round the outside of the 
gla-s, very small anode currents may be measured by an 
electrometer without any disturbance due to leakage. 

It is not necessary for ordinary wireless purposes that the 
valves should be constructed with the grid and plates either 
accurately circular in section or accurately centred with 
regard to the filament ; but for the purpose of calculation a 
svmmetrical system of electrodes is necessary and the M.T.5 
valve used in most of the experiments was specially selected. 
All the numerical results to be quoted were obtained from 
this valve. There is no difficulty in getting the short-wave 
oscillations with many types of hard valve, the French type 
produces them quite easily, but the chief reason for selecting 
the Marconi M.T.5 type was that the electrostatic field 
between the square-mesh grid and the plate approximates 

M2 



164 Messrs. E. W. B. Gill and J. H. Morrell on Short 

much more closely to the calculable field between two co- 
axial cylinders than does the field in the French type, wher^ 
the grid is a spiral coil of fine wire. The diameter o£ grid 
used was 1 cm. and that of the plate was 2*5 cm. to an 
accuracy of about 5 per cent. 

4. The preliminary experiments were made with the 
apparatus arranged as in fig. 2. The valve is shown dia- 
grammatical ly : F is the filament, Gr the grid, P the plate, 
LL the Lecher wires, which were of copper wire each about 
850 cm. long and spaced 5 cm. apart. They were suspended 
about 200 cm. above the floor from insulators secured to the 
walls at each end, and from one end were leads about 70 cm. 
long to the grid and plate respectively. The bridge consisted 
of two equal condensers, C, 0, joined through the heater-coil 
of a Paul thenno-j unction, T. The outer plates were fitted 



Fiff. 2. 




with contacts to slide along the Lecher wires. The capacity 
of these condensers is unimportant, provided it is large com- 
pared with the capacity of the valve. In practice, the 
capacities were of the order of 1 milli-microfarad. The 
terminals of the thermo-junction were connected to a gal- 
vanometer by two long leads, which are not shown. The 
sliding contacts were also connected to the negative side of 
the filament-heating battery B, that on the grid-wire through 
a high-tension battery V, and that on the plate-wire through 
a potentiometer S. which could raise the potential of the 
plate +6 volts above the negative end of the filament. 

Two sensitive milliammeters, A, A, gave the steady currents 
through the valve to the grid and filament respectively. A. 
rheostat, R, controlled the filament-heating. In all cases 
potentials are measured with regard to the negative end of 
the filament. 

electrons set free at the 



With this arrangement th< 



Electric Waves obtained by Valves.- 165 

filament move outwards under the positive voltage, V, oL' the 
grid, and a certain number go direct to the grid and arc 
collected there, the remainder pass through the grid, and, if 
the potential of the plate is just less than that of the filament, 
they return to, and are finally collected on the grid. If, on 
the other hand, the plate potential is a little above that of the 
filament, a certain proportion of those getting through the 
grid reach the plate. If the plate potentiometer is now ad- 
justed till the plate current is just zero, and the bridge is 
moved along the wires, it will be found that with the bridge 
in certain regions a plate current appears. It was the 
appearance of this plate current which led Barkhausen to the 
discovery of the short waves. With the present apparatus 
these oscillations are also made apparent by the deflexion of 
the galvanometer attached to the thermo-j unction. The 
positions of the bridge at which the galvanometer gave a 
maximum deflexion were fairly sharply defined, and did 
not always coincide with the positions for maximum plate 
current. 

It is not necessary for the plate potential to be so adjusted 
that the plate current is just zero when oscillations are not 
occurring. The plate may be set at a considerable negative 
potential, or the plate voltage may be positive. It was found 
that for a given grid potential there is a certain plate potential 
at which the oscillating current through the thermo-j unction 
is a maximum. Also as the potential of the plate was 
increased, for plate potentials only slightly positive, if oscil- 
lations commence the plate current increases ; at a certain 
plate potential no change is noticed in the plate current ; 
and at higher potentials the plate current decreases. For 
the M.T.5 valve this critical potential was about 4- 2 volts, 
when the voltage drop down the filament due to the heating 
current was about 4 volts. 

In the first experiments with this apparatus the position of 
the bridge was varied and the current in the thermo-couple 
observed when the grid voltage V, the heating current, and 
the plate potential were all kept constant. 

The oscillating circuit consists of a condenser formed by 
the plate and grid of the valve, the distributed inductance 
and capacity of the Lecher wires up to the bridge, and the 
capacities C, C in series with the wires and with the short 
resistance of the thermal heater which connects them. Hence, 
if there is an optimum wave-length A corresponding to the 
grid voltnge V, and if, starting near the valve, the bridge is 
pushed along; the wires, maximum amplitude of oscillation 



166 Messrs. E. W. B. Gill and J. H. Morrell on Short 

should occur when the above circuit is tuned to X, 2\, 3X, 
etc., these positions being indicated by the deflexions of the 
galvanometer connected to the thermo-j unction. Moreover, 
the distances measured along the wires between successive 

positions of maximum oscillations should be equal to - , and 

all therefore should be equal. It was soon found that this 
simplicity was not attained, in certain cases equi-spaced 
positions were found, but in the majority of cases there were 
at least two sets of positions forming two series of equal 
spaces, which, as the spacing distance of the two sets was 
different, appeared to indicate two optimum wave-lengths. 

These effects are due to the different modes of oscillation 
of the system, and, according to the theory which we give 
below, a grid voltage V will, under suitable conditions, sustain 
oscillations of short wave-length between certain limits. Any 
mode of oscillation corresponding to a wave-length between 
these limits will be maintained. It was therefore desirable 
to arrange the apparatus so as to avoid these complications. 

5. The most obvious improvement was to give up the idea 
of finding the wave-lengths by moving the bridge, and to put 
the bridge and its leads at the far end of the parallel wires 
joined to the valve, and to measure the wave-lengths of the 
oscillations by means of a loosely coupled secondary circuit. 
The system of wires connected to the valve is thus fixed. A 
second pair of long Lecher wires were set up with a loop 
joining one end, and this loop was brought near the valve 
circuit. When the secondary is in tune with an oscillation 
in the primary the current in the primary is reduced. The 
deflexion of the galvanometer connected to the thermo- 
junction in the primary circuit may be reduced by 50 per 
cent, when the bridge in the secondary circuit is in the tuned 
position, and a movement of 0*5 cm. either way will restore 
the deflexion to its original value. The distances between 
the successive positions of the bridge on the secondary circuit, 
for which the deflexions of the galvanometer attached to the 
primary circuit are a minimum, are the same, and are equal 
to halt the wave-length of the oscillation in the primary 
circuit. All the wave-lengths quoted were measured on 
this form of wave-meter and may be taken as accurate to 
0*5 per cent. * 

With the condenser bridge and thermo-couple at the far end 
of the Lecher wires the filament was heated to give an 
emission of a few milliamperes (this is low r heating for an 

* Townsend and Morrell, Phil. Mag. Aug. 1921, pp. 266-268. 



Electric Waves obtained by Valvesj 167 

M.T.5 valve) and the grid voltage was raised by two volts at 
a time by means of batteries of small accumulators from 16 
volts to 120 volts, while the plate was kept about 2 volts 
positive, as this gave large deflexions. The corresponding 
galvanometer deflexions are shown in fig. 3. 

The deflexions are plotted against grid volts ; as a thermal 
detector was being used, the deflexions are proportional to 
the mean square of the oscillating current. 

The curve shows that oscillations are occurring over nearly 
the whole range, but that there are maxima for certain 
voltages— viz., 16, 24, 42, 58, 82, 114, approximately. The 
wave-lengths measured as above give from 16 to just below 
24 volts X586 cm., from 24 to 40 volts X451, and so on, the 
wave-lengths for successive portions of the curve being 366, 
307, 262, 233. These correspond to the free oscillations of 

Fiff. 3. 




the system, the wires of which were 850 cm. long with 
leads to the valve about 70 cm. long, with a slight addition 
tor the leads within the valve itself. 

The system of wires connected to the valve therefore 
present a selection of various modes of oscillation with wave- 
lengths 586, 451, 366, 307, etc., cm., from which the valve 
chooses the one suitable for the particular voltage V between 
the grid and plate — the sharp rises just before the various 
maxima showing that the system oscillates on the longer 
wave-lengths by preference. For each particular wave- 
length there is a certain grid voltage which gives the 
strongest oscillations when the heating current in the filament 
and the plate voltage is constant ; but the heating current 
and the potential of the plate relative to the filament both 
affect the optimum voltage for a given wave-length. In- 
creased emission has the same effect, but this effect depends 
on the degree of saturation of the emission current. 

In the preceding experiments the wave-lengths of the 
oscillations were measured with a constant heating current 



168 Messrs. E. W. B. Gill and J. H. Morrell on Short 

in the filament, but the current from the filament varied 
with the grid voltage. For the lower voltages all the 
electrons leaving the filament do not r^ach the grid space, 
some returning to the filament. For theoretical reasons it is 
more convenient to find the grid voltages which give the 
maximum amplitudes of oscillation on the various wave- 
lengths when the heating current is so adjusted that the 
same current flows from the filament to the grid space for all 
the voltages, the plate voltage being kept constant as before. 
The table below gives a set of experiments done under such 
conditions with an emission current of 6 milliamperes, and 
the plate at 1'3 volts positive to the filament. In column 1 
are given the wave-lengths \ in cms., in column 2 the grid 
volts V, which excite these wave-lengths most strongly, and 
in column 3 the product X 2 V: — 



X. 


V. 


X 2 Y. 


208 cm. 


156-5 


68xl0 5 


233 


122-5 


66 


262 


92-5 


64 


307 


68-5 ' 


64-5 


366 


505 


67-5 


451 


36-5 


74 



All these results, with the exception of the last, agree well 
with the relation X, 2 Y = const. 

It is not difficult to see why this agreement should be less 
exact as V decreases. The electrons concerned are not all 
moving under similar conditions. Owing to the voltage drop 
of the heating current down the filament, the field between 
filament and grid differs by about 4 volts for electrons starting 
from the extreme ends of the filament. And when V be- 
comes comparable to this 4 volts a disturbing factor is 
introduced. 

6. These experiments thus give the grid voltages which 
produce the strongest oscillations on certain definite wave- 
lengths determined by the particular length of wire used. 

To find the range of wave-lengths maintained by a given 
grid voltage a slightly different apparatus (fig. 4) was used. 
An adjustable circuit was constructed of two rods, and two 
telescopic tubes fitted over the rods, so that the effective 
lengths of the system could be varied by sliding the tubes 
over the rods* 

The condensers and thermo-jnnctions were attached at the 
endsX, X 1 of the rods, and the ends Y, Y 1 of the tubes were 
connected to the plate and grid of the valve respectively, the 
other connexions being as before. 



Electric Waves obtained by Valves. 



169 



For brevity, the adjustable circuit will be referred to as 
the rods. It is not possible to graduate the rods in wave- 
lengths as against extension of the arms, as this wave-length 



Fig-. 4. 




depends on the emission and on the plate volts. Thus with 
a fixed length of the arms and 44 volts between grid and 
plate : 

(1) With plate potential fixed. 

Emission 2*2 m.a. \ = 311 cm. 

6-8 306 

9-8 300 

(2) With emission constant at 5*2 m.a. 
Plate potential 1*2 volts. \ = 308 cm. 

2-4 314 

- Hence for a given setting X decreases as the emission rises, 
and increases as the plate voltage is increased. This is due 
to the fact that the plate and grid are not a potential node of 
the oscillating system, but are a variable distance from it 
depending on the alternating voltage necessary to sustain 
the oscillation, and this in turn depends on the emission and 
plate voltage. It is not, however, necessary to go further 
into this, as the wave-lengths were always found directly by 
a secondary circuit as in Paragraph 6, the rods being used 
as a convenient way of varying continuously the wave-length 
of tire system connected to the valve. With all the other 
factors fixed, the rods were pulled out a centimetre at a time 
and the oscillating current and wave-length recorded for 
each position. In one experiment the emission was 1*5 m.a.. 
the orrid potential 44 volts, the plate potential 1'8 volts, and 
oscillations were maintained from X = 320 cm. to X = 451cm. 
with a maximum oscillation about A, = 323 cm. It was 
always found that the maximum oscillation was close to the 
short-wave end of the rane^e. 



170 Messrs. E. W. B. Gill and J. H. Morrell on Short 

The effect of: (A) varying the emission current keeping 
the grid and plate voltages constant, and (B) varying the 
plate voltage keeping the emission current and potential 
between grid and plate constant was investigated with this 
apparatus. 

In (A) increased emission broadened the range and de- 
creased the wave-length of maximum oscillation. 

For example, with V^ p = 44 volts and ~V p f—l'2 volts the 
wave-length for best oscillation with total emission 7*0 m.a. 
was 295 cm. With total emission 10'6 m.a. it was 274 cm. 

In (B) increase of plate voltage increased the wave-length 
and also broadened the range. Thus with Y^ = 44 volts, 
and total emission 3*8 m.a. with Y p f=l'2 volts X = 298 cm., 
and for V^ = 3*0 volts \ = 321 cm. This last observation 
must not be confused with the case in which the potential of 
the plate is increased and that of the grid kept constant. In 
that case also, increase of plate potential increases the length 
of the strongest wave, as was observed by Barkhausen, who 
attributed all the difference in wave-length to the alteration 
in potential difference between plate and grid. This cannot 
be the whole of the explanation, for, as stated above, similar 
results may be obtained by raising both plate and grid 
equally with respect to the filament. 

7. A simple theory to account for the maintenance of the 
oscillations can be worked out by making some simplifying 
assumptions ; but a general theory will not be attempted, 
partly because the resistance of the oscillating circuits used 
was unknown and partly because if the assumptions are not 
made the calculations become extremely complicated. 

These assumptions are : — 

A. That the grid and plate can be regarded as forming a 

parallel plate condenser. 

B. That, of the electrons which leave the filament, a fixed 

small proportion pass through the grid in a uniform 
stream, and that each electron on passing through the 
grid has the same velocity. 

C. That the electrons which return to the grid from the 

plate side are nearly all collected directly on it, i. e., 
only a few pass through on the return journey. 

D. That the oscillating potential differences are small com- 

pared with the fixed potential differences employed. 

It is also assumed that the pressure of the gas inside the 
valve is so low that the number of collisions between electrons 
and gas molecules is negligible — this is certainly true for the 
valves used. 



Electric Waves obtained by Valves. 171 

With these assumptions we shall only attempt to show that 
an oscillation can be maintained of about the right order of 
wave-length. 

The principle involved is the following : — 

Suppose the filament and plate are at zero potential and 
the grid at + V ; then the electrons from the filament which 
pass through the grid with a velocity v due to the potential 
V come to rest at the surface of the plate and return to the 
grid, which they again reach with velocity v. In the space 
between the grid and the plate the total work done by the 
fixed potential V on the electrons which move in this space 
is zero, all the work having been done between the filament 
and grid. 

If now superposed on the fixed potentials there is an 
alternating potential V s'mpt between grid and plate due to 
oscillations, the work done by the potential V sinp£ on the 
electrons is not necessarily zero. If the work is positive the 
electrons are abstracting energy from the oscillating system, 
and the average velocity with which the electrons hit the 
grid is increased ; the oscillations cannot in this case be 
sustained by the movement of the electrons. But if the work 
is negative the electrons are giving energy to the oscillating 
system, and if the rate at which this energy is given is at 
least equal to the rate of dissipation of energy in the oscil- 
latory circuit by resistance, radiation, or dielectric loss, the 
valve will maintain the oscillations. The average velocity 
with which the electrons hit the grid is in this case less than 
the velocity r due to the potential V which they acquire 
between fliament and grid, and hence the energy put into the 
system from the battery Y is not all used in heating the grid 
but part is turned into energy of oscillations. 

The above argument is not affected if, in consequence of 
the oscillation, some of the electrons are collected on the 
plate. In all cases, provided the total work dune per oscilla- 
tion by the alternating field is negative, an oscillation can 
be sustained if the dissipation of energy in the oscillatory 
circuit is not large. 

8. The particular case in which the filament and the plate 
are at the same potential when there are no oscillations may 
be considered first. Let V be the potential above the plate 
of the grid and cl the distance between them. 

"When there are no oscillations the electron passes the grid 

with velocity v\ = \/ — V and is then subject to a constant 
retardation/, which brings it to rest just at the plate. If T 



172 Messrs. E. W. B. Gill and J. H. Morrell on Short 

is the time the electron takes to pass from grid to plate, 
v = fT. A further interval of time T brings the electron back 
to the grid with velocity v. 

Assume now that superposed on the fixed potentials is an 
alternating potential V sin pt between plate and grid; the 
electric force due to this in the space between plate and grid 

y 

is ~ sin pt, and if — e is the charge on an electron the corre- 
spending force on it is — -—sin pt towards the plate. 

Since V is taken to be very, small compared with V the 
motion of the electron may to a first approximation be taken 
as determined solely by V, i. e., its time across is T and 
retardation /. 

The work done by V sinjt?£ depends upon the time t at 
which the electron passes the grid, and for a particular value 

of t the work is equal to l •— -psin ptdx. The axis of x 

being perpendicular to plate and grid and x = being on the 
grid. 

But the velocity at time t is 

JW-/(«-i„)=/T -/(*-<„), 

and the above work reduces to 



J 



fe — j (T -f t — t) sin pt dt, 

to 



(1) 



which finally gives ; 

Work on electron going from grid to plate = 

2eY / r Y oospto sinpt — sinpT-ht \ 

similarly, the work done on the same electron as it returns 
from plate to grid comes out as 

-rpr{-cospt + 2L+ — C-s p2 —)■ • 12) 

Thus the velocities of the electrons on their arrival at the 
plate or on their return to the grid depends on t , that is, on 
the value of V sinp£ at the instant they pass through the 
grid. Assuming a constant stream of electrons through the 
grid, it is easily seen by integrating (1) for values of t 

between and — that the total work done per period is 
p 



Electric Waves obtained by Valves. 173 

and similarly for (2). Hence, if all the electrons returned 
to the grid an oscillation would not be maintained. The 
possibility of a maintained oscillation depends in this case 
on the fact that in each oscillation a certain group of the 
electrons are collected on the plate and the integral of (2) 
does not in consequence include all the values of t between 

2-7T 

and - and its value is not therefore zero, but may be 

P 

negative. 

The first step is, therefore, to find which electrons reach 
the plate. When there are no oscillations the electrons have 
sufficient energy on passing the grid to just take them to the 
plate against the potential V, and if therefore any extra 
work is done on them they will be collected on the plate, 
but if the work is negative they will fall shore of the plate 
and return to the grid. Expression (1) shows that all the 
electrons which pass through the grid at times t , such that 

Tcos/>^ sin pt — sinp(T + '. . . .,, , , 

; + — —- 1 — is positive, will reach the 

• P l r 

plate, while chose for which it is negative just fail to reach 

the plate and return to the grid. 

Of the electrons then which pass the grid half go on to 
the plate and half return to the grid, the electrons running 
to the plate for a time equal to ir\p (half the periodic time 
of the oscillation) and then running back to the grid for 
time 7r/p and. so on. But the total work done by the 
oscillating potential on the two halves as they go from 
grid to plate is zero : and therefore the net work done 
is the work done on the return journey on the half which 
returns to the grid. 

To find therefore if an oscillation whose periodic time is 
2irlp and amplitude V can be sustained by a grid voltage V 
it is necessary first to find the time T which the electrons 
take to pass Irom the grid to the plate under the field due 
to V alone, next to find from equation (1) the values of t Q 
for those electrons which return to the grid when the 
system is oscillating, and, finally, by taking the mean value 
of expression (2) for these values of t and, knowing the 
emission current, to find the total work done per second by 
the oscillating potential. If this work is negative and at 
least equal to the dissipation loss per second, the oscillation 
will be maintained. 

A table of calculated approximate results is given below 
for various values of the ratio T : l/p. The second column 
gives the values of pt for the electrons which return 



Values of pt 

for electrons which 

return to grid. 


Work. 


Y-4 


165° to 345° 


negligible. 


1040 cm. 


150° „ 330° 


-•47 


520 „ 


135° „ 315° 


-•85 


347 ., 


120° „ 300° 


-•36 


260 „ 


90° „ 180° 


-•32 


130 , 



174 .Messrs. E. W. B. Gill and J. H. Morrell on Short 

to the grid in each oscillation, the third gives the total 
work done per second by the oscillating potential in 
arbitrary units for a fixed value of V and of emission, 
and the fourth gives the corresponding wave-lengths for 
the particular value, V = 44, for which T — 4*3 x 10"^ second 
(see next paragraph) : — 



r ~ 4T 

7T 

3;r 

7T 

2tt 
P = y 

In all these cases the work is negative and oscillations 
can be theoretically sustained, though in practice the 
dissipation losses in the oscillatory circuits are such that 
generally only those wave-lengths corresponding to the 
larger values of the work exist. 

For all values of the ratio T : ljp outside the range of 
the table the work is small. The larger the work the 
greater the amplitude of oscillation that will be sustained, 
and the periodic time 2ir/p of the oscillation of maximum 
amplitude for a given value of T is seen to be in the region 
of § T. In general, the wave-length corresponding to this 
will be sustained, and also a certain range of wave-lengths 
on both sides of it, the limits of the range being determined 
by the dissipation losses in the. oscillatory circuit. The 
theoretical result is in good agreement with several of the 
experimental results of paragraph 6. 

In the particular case recorded there for V = 44 the 
range of wave-lengths sustained was from 320 to 451 cm., 
with a maximum amplitude for 323 cm. Increase of 
emission broadened the range of wave-lengths sustained, 
which is in accordance with the fact that for a given V 
the work put into the oscillatory system is proportional 
to the emission current. 

The fact that the wave-length of maximum amplitude 
of oscillation was near to the short wave-length end of 
the range also agrees with the calculated fact that the 



Electric Waves obtained by Valves. 175 

work done falls off much more rapidly on the short-wave 
side of the maximum than on the long-wave side. 

9. To calculate the time T an electron from the filament 
takes to go from the grid to the plate when the grid potential 
is V volts above both filament and plate it is not necessary 
to assume the grid and plate to be parallel, but they may be 
taken, as they actually are, to be concentric cylinders of 

radii a, b. 

The retarding force on the electron when it is at distance v 

k b 

from the axis is -, where & = V/log e -. 
t a 

The equation of motion is therefore 



l 2 r - eh 



m 



which oives when integrated twice, remembering that 
dr 



= 0, when r = b, 
at 



\/jv 1o 4J 



V 



log 



T = b \ / -w log, - ] e~ x \lx. 



In the actual valve used a = '5 cm., 6 = 1*25 cm., and, taking 
-'= 5*3 x 10 17 E.S. units and measuring V in volts, 

711 

T = - r= second, 

w 

the accuracy of this being limited by the accuracy to 
which a and b are known and probably from 5 to 10 per cent. 

The wave-length for any relation between T and 1/p can 
now be at once calculated. If p = n7r/T, the time of one 
oscillation is 2ir/p or 2T/n and the wave-length in cm. is 
6xl0 10 T/>?,. 

The simple theory shows that for the oscillations of 
maximum amplitude pT has a certain value about 37r/4. But 

Tec , andXx — , and hence the connexion between the 

v 7 V P 

grid voltage V and the wave-length X of maximum 
oscillation is X 2 V = constant. 

10. The theory is thus in good general agreement with 
the experimental results, but there is one fact unaccounted 
for — that being the variation in the wave-length of the 



176 Messrs. E. W. B. Grill and J. H. Morrell on Short 

oscillation o£ maximum amplitude, for a fixed potential 
between grid and plate, when either the emission is altered 
or the plate potential is slightly altered with respect to the 
filament. 

There is also a special case, which is forming the subject 
of a separate investigation, in which, when the plate is 
very negative (40 volts or so) with respect to the filament, 
oscillations can still be produced, but without any current 
reaching the plate at all. These oscillations are, however, 
very much weaker and more difficult to produce than those 
dealt with in our experiments. 

The simple theory which depends on the collection on the 
plate in each oscillation of a group of electrons will obviously 
not account for this special case. 

The explanation of the above considerations is to be looked 
for in the assumptions made in the simple theory. The first 
assumption that the grid and plate could be regarded as 
parallel is not important, as the field between cylinders of 
the size of the grid and plate used is not far from uniform. 
(It will be noted that the value of T was calculated for the 
valve used by taking the field between cylinders.) The only 
difference between cylinders and parallel plates on the simple 
theory would be to make the ratio T to 1/p for maximum 
oscillation slightly different. But the second assumption 
that the electrons pass the grid in a constant stream all 
having the same velocity requires more careful examination. 
This velocity is not actually the same for two reasons : 

A. Because there will be alternating potentials between 

the filament and the grid which set up a velocity 
distribution at the grid. 

B. Because of the voltage drop of the heating current 

down the filament. 

In the usual methods of producing oscillations by means 
of valves the alternating potential of (A) is most important, 
as it controls the whole action of the valve, but in our 
experiments it is only of secondary importance. The 
alternating potentials induced between filament and grid 
are smaller than those between grid and plate, and the 
major part of the work done by the alternating field on 
the electrons, which is what determines whether the 
electrons reach the plate or not, is done between grid 
and plate, and it is therefore nearly correct to say that 
all the electrons passing the grid at times t , such that 
expression (1) is positive, reach the plate. 

In the extreme case, however, when the plate is so 



Electric Waves obtained by Valves, 177 

negative that the system is unable to oscillate by the method 
of: driving groups of electrons on to the plate, the oscillation 
is almost certainly due to a velocity distribution at the 
grid, as this means that the electrons do not pass the grid in 
a uniform stream, and allows the integral of expression (2) 
to be finite and not zero, as it normally is when all the 
electrons return to the grid. 

The comparative weakness of the oscillations in this case 
shows that the electrons have all nearly the same velocity 
when passing the grid. 

The simple theory should therefore be in agreement with 
the observed facts, as it is when the oscillations are mainly 
due to the collection by the plate, but as the plate is made 
more negative with respect to the filament the velocity dis- 
tribution at the grid becomes more important and the simple 
theory is less accurate. 

The velocity distribution at the grid will also bo affected 
by the emission, as this varies the space charge round the 
filament — this affecting the time the electrons take to pass 
from, the filament to grid, — and this in turn varies the small 
effect of the alternating field in this space. 

The effect of the voltage drop of about 4 volts down the 
filament is that, instead of dealing with one stream in 
the field due to the grid being charged to V volts, there 
are a series of streams moving under potentials varying 
from V to V — 4 (V being the potential difference between 
the grid and the negative end of the filament). The number 
of electrons in the various streams varies from a maximum 
number corresponding to V — 2, the middle of the filament 
being the hottest. The emission falls off equally on both 
sides of this middle point. 

In the general case, when the plate is slightly positive with 
regard to the negative end of the filament when there are no 
oscillations, some of the streams reach the plate and the 
remainder approach it closely, but to varying distances. 

If oscillations commence some of these latter streams are 
periodically diverted to the plate, while in the other half 
oscillations some of the former are diverted off. 

Thus all the streams concerned maintain the oscillation as 
in the simple theory, and unless Vis small the wave-lengths 
they each maintain best are nearly the same, so that the 
combined effect differs little from that of a single stream 
moving under potential V. 

The question of whether the mean plate current rises or 
falls when oscillations begin depends on whether the average 
Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. N 



178 Short Electric Waives obtained by Valves. 

density of the streams diverted to the plate exceeds or is less 
than that of those diverted from it. 

If the plate is at the same potential as the centre of the 
filament, no change should therefore occur. If it is above 
this the current should drop, and if below the current should 
rise, which is in agreement with the results in paragraph 4. 

In conclusion, we should like to express our thanks to 
Professor Townsend, who has assisted us with much valuable 
advice and criticism. 



Note on the Determination of Ionizing Potentials. 

In the experiments above described, the electric fields in 
the spaces between the grid, filament, and plate are similar 
to those used in experiments on the determination of critical 
potentials when a small quantity of gas is introduced. 

In the latter experiments variations in the plate current 
are observed as the grid potential is raised, and at certain 
potentials of the grid abnormal variations in the plate current 
are observed which are interpreted as indicating certain 
critical potentials, characteristic of the molecules of the 
gas. 

The experiments which are here described show that 
abnormal variations in the plate current are to be expected, 
due to oscillations which may be maintained where large 
currents of the order of a milliampere flow from the filament 
towards the grid when the gas is at a very low pressure. The 
effect of varying the potential of the grid is clearly shown by 
the curve of fig. 3. 

In all the ordinary methods of wiring the valve to the 
cells and galvanometers or electrometers, the system seems 
to be as often in a state of oscillation as not. 

Even if the plate is so negative that the oscillations are of 
the weaker type which do not affect the plate current, the 
difficulty arises that the oscillations superimpose an unknown 
potential difference between filament and grid, and the 
electrons are not moving under the fixed field alone. 

It is necessary therefore, in order to obtain reliable values 
of the critical potential, to take precautions to prevent 
oscillation, which is best done by using emission currents 
much less than a milliampere. 



[ !''•» J 



XTI. Pseudo- Regular Precession, 
By Sir G. Greenhill *. 

I^HIS is the gyroscopic motion described and illustrated 
in Klein-Sommerfeld's Kreisel-Tkeorie, p. 209, where 
a spinning top appears at first sight to be moving steadily in 
uniform precession at a constant angle with the vertical, but 
on closer scrutiny the axle is seen to be describing a crinkled 
curve of small loops or waves; so that in this penultimate 
state a realisation is obtained of a motion expressible by a 
function which does not possess a differential coefficient, 
a paradox fascinating to a certain school of pure mathe- 
maticians. 

A pseudo-regular precession, although invisible, would 
not feel impalpable to the analytical thumb passed over it, 
which would detect a roughness. But in regular precession 
the curve would feel quite smooth. 

In the investigation the axle may first be supposed moving 
in perfect steadiness with no trepidation or nutation ; and 
then to receive a small impulse, blow or couple, giving rise 
to the pseudo-regular precession visible to the eye. 

1. Begin with a rapid spin about the axle, held fixed at a 
constant inclination, taken at first as horizontal for simplicity, 
in fig. 1. 

Fisr. I. 




For visible experimental illustration, it is convenient to 
take a large (52-iiuh) bicycle wheel, mounted on a spindle 
with ball bearings, and to prolong the spindle by sere wing- 
on a stalk, at one end or both. 

The end of the stalk may be supported on the hand and 
the wheel set in rotation by a swirl ; the hand accompanies 
the wheel in the precession ; or else the point may be placed 
in a cup fixed on the floor. 

When the axle is released from rest, it will start from a 
cusp and sink down, then rise up again in a regular series of 
loops or festoons ; so that to secure the uniform precession 

* Communicated by the Author. 

N 2 



180 Sir G. Greenhill 



on 



an impulse couple must be applied, given by a horizontal 
tap of appropriate amount. 

The word moment or momentum is of such frequent 
occurrence in dynamical theory that we prefer to replace it 
by impulse instead of momentum, linear or rotational. 

Representing then the impulse (rotational) CR due to the 
rotation R about the axle by the vector OC, the impulse 
required to start the rotation from rest, or reversed to stop 
it again, the axle OG in steady motion will move round the 
vertical OG at a constant (horizontal) inclination, with pre- 
cession /j,, such that the vector velocity of K, the end of the 
resultant impulse OK, is equal to the impressed couple of 
gravity ; then CR{A=gM.h, M.h denoting the preponderance 
or first momeni about 0. 

This result, true accurately when the axle OC is hori- 
zontal (fig. 1), is obtained at any other inclination 6 of the 
axle with the vertical (nadir or zenith, figs. 2, 3) in 
the elementary Kindergarten treatment, where the top is 
supposed spinning so fast that the deviation is insensible 
of axial impulse OC and resultant impulse OK, and then 
the velocity of C may be equated to the gravity couple, 
making CR//, sin 0=gM.h sin 6, as before, when the axle was 
horizontal, on dividing out sin 6. 

2. Hanging down inert, vertically from in fig. 2, the top 
forms a compound pendulum of S.E.P.L. (simple-equivalent- 
pendulum-length) 0P = £ = A/M/t, A denoting the trans- 
verse-diametral M.I. (moment of inertia) at 0, and as 
above the axial M.I. ; and in small invisible oscillation in a 
plane, the top will swing as a pendulum, and beat njir times 
a second, where n 2 =g/l, An 2 =gMh, or make a swing or 
beat in ir/n seconds ; MA may be called the preponderance 
about 0. 

Falling down from rest from" the upward vertical position, 
the top will have acquired the angular velocity co in the 
lowest position, such that by the Energy-Principle, 

iAo) 2 = %M/>, 

and the equivalent rotational impulse Aa) = 2A??, while 
An 2 =gMh is the equivalent of P in the Kreisel-Theorie. 

The impulse 2An = 2 ^(gMhA) is a dynamical constant of 
the top, and to a geometrical scale may be represented by a 
length Jc, in addition to the S.E.P.L. 0P = Z measured along 
OC ; and then if in any assigned top motion the constant 
impulse component G and CR, about the vertical OG and 



Pseudo-Regular Precession. 181 

the axle OC, is represented to the same scale by d and D', 

a, a' = G, cr = m' 

/t ' 2An ' a 

in Darboux's notation (a different use of A from that 
employed above). 

Time can be reckoned in the pendulum beat, irjn seconds ; 
and the relation, CB. /n=gM.h = An 2 , can be written 

jji _ A n 

n ~CR ; 

or expressed in words, the number of beats per circuit of the 
axle is C/A times the number of revolutions of the top per 
double beat. 

The resultant impulse vector being OK, the component 
perpendicular to the axle, if horizontal, as in fig. 1, is 

A _ (?MAA _ A 2 n 2 
^"""CfT" CR' 

or to the geometrical scale, OC.CK = i& 2 , in the steady, 
regular precession. 

For brevity we are allowed to assume tacitly the geo- 
metrical scale, and to replace any dynamical quantity in an 
equation by its vector length, such as the axial impulse CR 
by the vector length OC, or a'. 

3. To change this steady motion of the axle into a 
penultimate pseudo-regular precession, another impulse is 
applied about a vertical axis, supplied by a horizontal tap 
on the axle perpendicular to the plane OCK, in fig. 1. 

This will cause CK to grow to CK 3 , and the resultant 
impulse to change from OK to 0K 3 ; and to make the 
pseudo-regular precession advance through a series of cusps, 
we find that KK 3 = CK, and the axle rises from OC to OC 2 
at an angle 2 with the upward vertical, zenith ; where C 2 
reaches the level GK 3 of K 3 ; and here # 3 , the inclination in 
the lowest position, is \tt. 

By a general dynamical principle 

OK 3 2 -OK 2 2 =i* 2 (cos<9 2 -cos0 3 ) 

= 2OC.CK(cos0 2 -cos6> 3 ). 

For in the general unsteady motion of the axle of a top, 
where the inclination 6 is varying, a new component KH is 
added to the impulse OK perpendicular to the vertical plane 



182 



Sir G. Greenhill 
dO 



GOC, of magnitude Ay-, and then the resultant impulse 

do 

vector OH describes a curve in a horizontal plane GHK, 
with velocity equal to the gravity couple gMh sin 6. 

Fig. 2. 




N' 

The radial velocity of H in the horizontal plane is then 

-t— < = gWi sin 6 cos GHK = gWi sin 6-^ , 

and integrating, 

i GH 2 = ?MAA(E - cos 6) = A 2 n 2 (.E - cos 0) 
= iP(E— costf), iOH 2 = iA: 2 (F-cos^), 

to a geometrical scale, with E, F dynamical constants. 

The component HK is zero in the upper and lower position, 
where = 6 2 , Z . 

In the general unsteady motion of the top, the impulse 

vector moves from OK to OH with KH = Ay- ; and with 

GC 2 
sin 2 0' 

OG 2 -2OG.OOcos0 + OC 2 
sin 2 



KH 2 =OH 2 -OK 2 = OH 2 
(A^Y=2AV(F-cos0)- 



Pseudo- Reg alar Precession. 183 

or with cos = :, and replacing OG, 00 for dynamical 
homogeneity by Darboux's 2A/j, 2AA', 

^y=2;i 2 (F-^(l-^)-4(A 2 -2/Ji'cos(9+A' 2 )==2n 2 Z, 

thus defining z as an elliptic function of t. 
Resolved into factors, we write 

r L — z x — z. z 2 — z. z — z s , 

in the sequence oo >c x > 1 >z 2 >z> c 3 > — 1 ; and then 

z = z 2 sn 2 \mt + z* 6 en 2 ^mt, m 2 = 2n 2 (^ 1 — %). 

4. Here with 00 horizontal in fig. 1, cos #3=0, 

OK 3 2 -00 2 2 =G 3 K s 2 -G 3 K 2 2 =CK 3 2 

= 200 . OK. cos0 2 = 2CK, . OK, CK 3 = 20K. 

A greater impulse would make the cusps open out into 
loops in the pseudo-regular precession; but the cusps would 
be blunted into waves if the impulse was reduced. 

Reverse this tap, and K is brought back again to 0, and 
the axle would fall as at first from a cusp and rise again. 

In the first cusp motion where the axle rises to a series of 
cusps and sinks again to the horizontal, the motion is found 
to be pseudo-elliptic and can be expressed in a finite form, 

sin 6 exp (yjr — lit) i 

= \/(l — cos# ? cos#) +i >/(cos # 2 eos# — cos 2 0), 

connecting azimuth i/r with 0, the inclination to the zenith. 
The verification is left as an exercise. Here h = h' cos# 2 . 

In the second cusp motion, where the axle is horizontal 
and falls from a cusp, and then sinks down to an angle 6$ 
with the downward vertical, nadir, the (yjr, 0) motion is not 
pseudo-elliptic ; but azimuth yjr and hour angle <f> change 
place (0, (j>, ijr the Eulerian angles), and 

sin 6 exp (cft — h't) i= V (sec 6 3 — cos 3 . cos 6) 

+ zV (cos # 3 — cos 6 . cos # + sec # 3 ), 

OR 
where Darboux's h'= ^— = i/(^ . sec S — cos 6 Z ) changes 

place with h, or d' with ~d ; h = h' cos # 2 and B now zero. 

But an interchange again of </>, ijr will give the (\jr, 6) 
motion of a non-spinning gyroscopic wheel, or spherical 



184 Sir G. Greenhill on 

pendulum, with h', CR and ~d' zero, in which the axle is 

projected horizontally, with angular velocity 2 h = sin 2 $-—, 

and sinks down to an angle 3 with the nadir, rising up 
again to the horizontal, and this makes 

}i—n V{i£ . sec 6 3 — cos Z ). 

The motion can be shown with a plummet on a thread, say 
about 10 inches long, to beat as a pendulum twice a second, 
a double beat period of one second ; whirled round swiftly, 
the thread rising to the horizontal position, and sinking down 
again periodically. 

Then we find 

v 2 2 = U 2 h 2 = 2gl(sec #3— cos 3 ), v% 2 = 2gl sec S , 

and in the conical pendulum, at angle # 3 , v 2 = ^v 2 2 . 
The apsidal angle is found to be 

^ = l7T+K i /(l-2« 2 ) -*7T(l-3 COS 2 8 ) 

as the plummet is whirled round faster. 

5. But next suppose the axle OC is held at an angle 6 
with the zenith, the wheel spun with impulse CR = OC, and 
then released, in fig. 3. 

The axle will start from a cusp, at 6 = 6 2 , and the motion 
in general is not expressible in finite terms as pseudo-elliptic ; 
but it will represent a gravity brachistochrone on a sphere. 

To make the axle move steadily at the inclination 6 with 
constant precession /*, the impulse vector CK is applied 
perpendicular to the axle, such that yu, sin 6 being the com- 
ponent rotation of the wheel about the axis OK' perpen- 
dicular to OC, CK = A/isin# (the inertia of the stalk being 
ignored), MO = Ayu,cos#, MK = A/*, KM drawn vertical to 
meet OC in M, with the condition 

GK . //,= gravity couple =gMh sin 6 — An 2 sin 0, 

0M= GK : Arf 
sin 8 /jl 

then the geometrical relation CM-f MO = OC becomes 

A?? 2 
Ayucostf-f — =CR, 
A 6 
the condition for steady motion. Also 

OM.MC=AVcos<9 = fPcos<9, OM.MK==JP, 

so that K lies on this hyperbola with asymptotes OC, OG. 



Pseudo- Regular Precession. 185 

Or the position of M is determined by drawing QQ' 
parallel to OC, to cut the circle on the diameter OC in 
Q,Q', when MQ 2 = ±Z: 2 cos0. 

This is for the small value of /-t in fig. 3, and the quiet 
precession when M is taken close to C, the other point Q' 
will determine a motion where the precession //, is swift and 
the motion violent. 

Fig. 3. 




OC, = 20M. 



MK' = 2MK. 



e 



Spinning upright with cos 0=1, fi-\ = -^ R ; and the 

fx A. 

two values of fi will give the independent normal invisible 

n 2 
circling of the axle round the vertical ; there will be ___ 

rapid beats for one slow beat of the axle. P 

The general slight oscillation will be compounded of these 

two circlings, adding up to an epicyclic motion of the axle, 

a result obtained in this manner without any appeal to 

approximation. 

In fig. 2, the axle is pointing downward at *an angle 6 

with the nadir ; here the slow precession of K is retrograde, 

but in the swift violent motion of the associated K', the 

precession is direct. 



186 Sir G. Greenhill on 

6. Next to make the axle rise to a cusp on d = 2 from 
= 6 Z in the penultimate pseudo-regular precession, the 
impulse applied is KK 3 , to make the axle come to rest on 
the horizontal G 2 K 3 in K 2 , C 2 , in fig. 3. 

Then in the general formula, or on the figure, with CD 
the perpendicular on K 3 M 3 , 



G 2 K 3 2 -G 2 K 2 2 = OK 3 2 -OK 2 2 = CK 3 2 -:P 2 (cos6> 2 -cos^ 
or CK 3 2 = K 3 D . K 3 M 3 =2 OM ,MK^, 

and producing MK double length to K', 



K 3 M 3 . 0C=20M . MK = OM . MK', 

implying that if OCLN' is the parallelogram on OC, the 
diagonal OL will cut MK' in L 3 such that ML 3 = M 3 K 3 , and 
K 3 is determined by drawing L 3 K 3 parallel to 00, cutting 
off the length CK 3 on OK, in fig. 3. 

7. If the impulse is applied about the axle of the top, to 
increase 00 to 00 3 , and make the axle rise to a cusp in 
fig. 3, 

OC 8 .K 3 M 3 = 20M.MK, with MK = K 8 M„ 

so that OC 3 = 20M. 

Thus the axle will rise from the horizontal in fig. 1 to a 
cusp by the application of an axial impulse CC 3 = 0C. 

8. The impulse might be applied about a vertical axis to 
the steady motion, making K rise vertically to K 3 ; and 
then in a cusped motion, with 00 changed to 0C 3 , and rising 
to 0C 2 at K 2 on the level of G 3 K 3 in fig. 4 ? 

G 3 K 3 2 -G 3 K> = OK 3 2 -OK 2 2 = C 3 K 3 2 =MK 3 . K 3 D 

= 20M . MK(cos0 2 -cos0 3 ) = 2OM . MK^ D , 

MK 8 . OG 3 = 20M . MK = 0M . MK', 

MK 3 _ MC 3 _ OM 
MK'~M0'~00 3 ' 

dropping the perpendiculars K 3 3 , K'C on 00 ; and drawing 
the circle on the diameter OC, with ordinate MQ', 

C 3 M . C 3 = 0M . M0' = MQ' 2 . 



Pseudo-Regular Precession . 
Then if L is the midpoint of OM, 

C 3 M . C 3 = LC 3 2 -LM 2 = LQ /2 , LC 3 = LQ', 
to determine C 3 . 

Fig. 4. 



187 




MK' = 2MK. 

MC = 2MC. 



L mid-point of OM. 
LC 3 = LQ'. 



9. Applied about a horizontal axis in the vertical plane 
GOC, the impulse will make K move horizontally to K 3 , and 
the axle rise to a cusp at K 2 , C 2 , in fig. 4, if 

C 3 K 3 2 = M 3 K 3 . K 3 D = 2 0M .MK^, 



M 3 K 8 .OC 3 = 20M.MK. 



M,K,_FC, OM 
MK ~ FC ~ OC 3 ' 

FO, . C 3 0=20M . FC -2GK . KF = G / K' . K'F', 

thence a geometrical construction may be devised for the 
determination of C 3 and K 3 . 

10. When the impulse is applied in the vertical plane 
GOC, the impulse vector starts out of the plane, from OK 
to OH, and K moves to H perpendicular to the plane GOC. 



188 Sir G. GreenhiU on 

The axle then oscillates between 6 2 and # 3 , 

e,<e<e % , kh=aJ, 



GH 2 = KH 2 + GK 2 = KH 2 + ( 



OC-OGcos<?\ 2 



CH 2 = KH 2 + CK 2 ^KH 2 + ( OG - 00 /°^ ) : 

\ sin / 



sin 6 J 
= 2OM.MK(E-cos0), 

f-OCcos<9\ 2 



= 2OM.MK(D-cos0). 



Fig. 5. 




When the axle rises to a cusp, rises to 2 , K 2 at the 
level of G, where in fig. 5 

KH = 0, OG-OCrcos0 2 = O, D=cos0 2 , 

OH 2 = 2 OM . MK (cos <9 2 - cos 6) 

= 20M . MK^5 = 2^ . KC 2 

KH 2 = CH 2 - CK 2 = (2~ -i) CK 2 , 
giving KH the impulse. 



Pseudo- Regular Precession. 189 

This caa be applied by hammering the rim o£ the bicycle 
wheel with a stick in a vertical blow at its highest point. 
Then 

OK3 2 -OK 2 2 = OK3 2 -OC3 2 = C 3 K3' 2 

= 2 OM . MK (cos 2 -co8 3 ), 

M3K3 . K 3 D 3 = 20M . MK l -^ } M 3 K 3 . 00=20M . MK, 

and this determines the level of AM 3 in fig. 5, and provides 
a geometrical construction for the position of 3 . 
For 

0M 3 . M 3 C 3 = OA sec 3 . M 3 K 3 cos 3 = OA . AG, 

so that, if B is taken in OG where AB = OA, 

4 0A . AG = OG 2 -BG 2 = OK 2 2 -K 2 B 2 ; 

and the circle centre K 2 {in d radius K 2 B will cut off a length 
OE on OK 2 = OF = 20M 3 , and so determine the direction of 
OM 3 C 3 , the axle in the lowest position. 
When OC is horizontal, as in fig. 1, 

cos# 3 = 5 CH 2 = 2OC.CKcos0 2 = 2CK 2 , KH = KC, 

as at first ; and OC = OM, M 3 K 3 = 2MK, 9 2 + 6, = tt, 
the axle oscillating to an equal angle above and below the 
horizontal. 

11. When the cusp motion is pseudo-regular and in small 
loops, it can be projected on the tangent cone of a sphere in 
a series of small hypocycloidal branches, and the motion is 
realised as discussed in the Principia, Book I, section X, 
when the tangent cone is developed into a plane and gravity 
radiates from a centre. 

With the axle horizontal, a necklace of brachistochrone 
cycloids is formed round the equator, with mean regular 
precession fi, fluctuating in azimuth between and 2fi, with 
azimuth interval 2^ = 2 Kk, tending as the rotation Rand 
axial impulse CR is increased to ttcos0 2 , in a zone above 
the equator of angular width ^it — # 2 , and area 2 7m 2 cos # 2 
on a sphere of radius a, and the number of cycloids in the 
necklace would be about 2sec# 2 . 

12. Even in the steadiest smoothest Regular Precession a 
close scrutiny will reveal under the slightest disturbance an 
almost invisible deviation from a perfect circular motion, in 
the shape of a progressive motion of an apse line, realised 



190 Sir G. G-reenhill on 

easily with the thread and plummet; utilised by Newton to 
illustrate the Evection of the Moon, 

Returning to the general unsteady motion of a top in § 3, 
and its vector impulse OH, the velocity of H imparted by 
the gravity couple An 2 sin is horizontal and along KH 
perpendicular to the plane GOC. This velocity is the rate 
of growth of KH, added to the velocity of K carried round 

by the plane GOC with angular velocity ~- ; so that, with 

KO = A sin 6 -^, and putting -,, — Q, 

An 2 sin0 = A-^ + GK-^ =A-^+ . a , 

dt 2 at at A sin 6 3 

. xl _ OC-OGcos0 rrn OG-OCcos<9 

with GK = ; — z , KC = — — -. — ^ , 

sm 6 sin 6 

, dGK KC dKG GK 

and 



dd "sinfl' dd sin0' 

obvious geometrically on fig. 3. 

Differentiating with respect to 0, with d0 = Qdt, 

1 d 2 Q 2 a GK 2 -fKC 2 ~GK.KCcos0 



= n? cos 



e-~ 



Q dt 2 ~ A 2 sin 2 

GC 2 -3GK.KCcos<9 



cos# 



A 2 sin 2 6 



-( 



2 , 30M . ON\ a OK 5 

n 2 H r^ I COS #■ 



A 2 



an exact equation. 

In a state of perfect Steady Motion of Regular Precession, 

Q and -7- are zero, and, in dynamical units, from § 5, 
OM.ON-AV, °M«i, gf-^-T' 

An />6 O.N /Z 2 Z' 

where X is the height of the equivalent conical pendulum, 

X=^ 2 = OA, l = g/n 2 = OY ; 

OC n u 

r-r— = - + - COS 6. 

An fM n 



OG 
An 


n A a 

= - COS04- ~, 
A 4 - W. 


GO 2 


OK 2 sin 2 (9 


An 2 


A 2 n 2 



in 2 6 /n 2 _ . , /j?\ . 9/i 
- T — = 2 4-2cos0+^ sm 2 0. 
?r \/r nV 



Pseudo-Regular Precession. 191 

Then in this Steady Motion, 

7^ -y^T -fwi — 0, where 

m 2 OK 2 . . \ „ 2 AP 2 

n 2= A^^ 4c0 ^ = 7- 2cOS ^ + X = OATOP^ 

£-2£ OA 

m 2 ~AP 2 UA * 

This result is exact and reached without any approxima- 
tion : and the slightest disturbance will give a nutation 
Q = Q cos {mt + e) , beating m/27r times a second, and the 
apsidal angle, from node to node, is 

Mr " 0P 

\ m AP 

In Darboux's representation of top-motion by a deformable 
articulated hyperboloid of the generating lines, the model is 
flattened into a rigid framework for Steady Motion ; and 
KM, KN produced to double length at S, S' will make the 
focal line SS' parallel to MN ; this will be revolved about 
the vertical line ON with constant angular velocity jul. The 
small nutation will be due to a slight play or baeklash in 
the frame. 

13. The same argument can be applied to the invisible 
oscillation of a Simple or Spherical Pendulum, or to the 
apsidal angle of a particle describing a horizontal circle on a 
smooth surface of revolution about a vertical axis. 

Taken as the axis Oy, the general equations of motion of 
the particle are 

1 dx 2 Idy 2 1 „d^ 2 , „ , . 

and x 2 -j- = K (impulse) ; 

d v d*\l/* 

so that, with -=-' =Q, and eliminating -j- , 

and differentiating with respect to #, dx = Qdt, Q C -~ = ~, 

dt\ 1+ dx 2 ) + ^ dx dx 2 x s +9 dx~ V > 
1 d^ ( djf\ dQdylij td 2 y\ 2 

Q dt 2 \ + dx 2 ) + dt dx dx 2 "*■ H \dx 2 ) 

~dy d 2 y , 3K 2 d 2 y A 
dx dx z x* J dor 
exact equations. 



192 Pseudo-Regular Precession. 

In a state of Steady Motion in a horizontal circle, 

Q = 0, ^=0, K=^ 2 , 

g = V?x d £- =V . NG, v*=^^g® d £ =g . NY? 

if the normal and tangent at P meet the axis Oy in G and V; 

and then with -f- = tan 6, 
ax 

oscillates between close limits #±/3, 

Q— (ot+ft) sin 2 \mt + {*—/3) cos 2 fynt, Q = m/3 sin mt ; 

and the particle beats mj2ir nutations per second, syn- 
chronizing with the beat of a simple pendulum of length 
\—g\m 2 , where 

A ~ VNG + dx 2 ) co * GV + PR' 

where PR is the semi-vertical chord of curvature upward of 
the profile curve of the surface. 

Then on a cone X = JGV ; and on a sphere or spherical 
pendulum, PR = NG, and to radius a, 

- =3 cos 6 + sec 6. . 
A, 

For a profile given by y = cx 7 \ 

d -l=n^, NG = ,^ = -, v 2 = 9 .NV = w, 
dx x dy ny * J J 

dx 2 x 2 dx 2 n + 2 

Thus on the surface of a free vortex, where the angular 
impulse xv is constant of all annular elements of liquid of 
the same volume, x 2 y is constant, and n— — 2, X = co . 

On a motor or bicycle track of this shape, the steering 
will be easy, and a change of place can be made without 
difficulty or danger, as with the annular elements of liquids 
in the vortex volume. 

A start is made with moderate velocity from the circum- 
ference of the track where the slope is slight, and the car 



The Binding of Electrons by Atoms. 193 

is steered with increasing velocity down towards the middle, 
where the cars can pass and repass without difficulty. 

To avoid a deep hole in the sink in the middle, the profile 
can change to the parabola of a forced vortex, where 

v= M , n = 2, y=^-, NG=p, A.=iGV = iSP. 

On a horizontal circle of this track of one lap to the mile, 
NP = 840 feet ; described in two minutes at 30 miles an 

NP 

hour, NV=60 feet, and cot 0= ^= = 14, a slope of 4°. 

Raise the speed to 60 miles an hour on this track, 
NP = 420, NV = 270, feet, and the slope is nearly 30°, the 
circuit of two laps to the mile made in 30 seconds. 

At a speed limit of 90 miles an hour, NP = 280, NV = 540, 
feet; round a circle of three laps to the mile, on a slope of 
over 62°. The surface could then change to a paraboloid, 
with a flat area in the middle, where a car could come to rest. 



XIII. The Binding of Electrons by Atoms. By J. W. 
Nicholson, F.R.S., Fellow of Balliol College, Oxford*. 

ACCORDING to the quantum theory of atomic structure 
and of the emission of line spectra, the paths of the 
electron in the atom vary according to the particular co- 
ordinates used in the process of quantizing the separate 
momenta. Thus in the simple case of a hydrogen atom, 
containing a nucleus and one electron, we may use either 
spherical polar or parabolic coordinates, and the admissible 
orbits are entirely different in the two cases. Yet the final 
values of the atomic energy are the same, and consequently 
each method yields the same theoretical spectrum. It has 
been suggested that there is in fact, in every case, only one 
type of coordinates which can be used, when all the modi- 
fying circumstances, such as the variation of the mass of the 
electron with speed, are taken into account. The only pro- 
blems yet solved are those in which the separation of 
variables, after the manner of Jacobi, can be effected, and 
the contention is in fact that there is, in every case, only one 
set of coordinates which allows this separation, when non- 
degenerate cases of the motion are discussed. 

But it is generally believed that the atomic energy is in 
all cases determinate and definite. We shall show, in the 

* Communicated by the Author. 
Phil. Mag. Ser. 6. Vol. 44. No. 259. July 1922. O 



194 Dr. J. W. Nicholson on the 

first place, that this conclusion requires modification when 
the path extends to infinity. The hyperbolic orbits of 
Epstein, which have been used extensively in the inter- 
pretation of certain groups of 7 rays associated with many 
of the chemical atoms, constitute an instance, and we shall 
show that they rest on a mathematical error, and that in fact 
it is not possible to preserve finite phase-integrals in the 
process of quantizing the momenta. In fact, it appears that 
the whole process is only applicable to finite paths, and gives 
no clue to the phenomena taking place during the binding 
of an electron which comes from a considerable distance. 

In another form, the question we propose is as to whether 
a hyperbolic path is possible in the same way as an elliptic 
one. Such would, of course, be characterized by a positive 
energy W. Certain available evidence of a simple kind, 
apparently not hitherto noticed, is in existence. For the 
existence of such paths involves the existence of parabolic 
paths, with W = 0. In passage from a stationary state of 
energy Wi (negative) to a parabolic path taking the electron 
outside the atom altogether, a quantity of energy Wi should 
be involved. Spectral lines given by 



hv = W 



Hi 



where W n corresponds to any one of the stationary states, 
should thus exist. In other words, the ' limits ' of spectral 
series should themselves be spectral lines. But there are 
two reasons why evidence on these lines cannot be decisive, 
especially when it is negative evidence. For in the first 
place, the values of W n determining the limits of series are 
of such magnitude that only for two or three, in any case, 
can the corresponding lines come into the visible spectrum, 
and with only hydrogen atoms and charged helium atoms 
to test, and enormous band spectra for both elements, the 
test cannot readily be applied. Moreover, the probability of 
an electron entering the atom in a parabolic rather than a 
hyperbolic path is so small that any resulting lines could 
hardly be expected to be of visible intensity under ordinary 
conditions. We consider, therefore, that the question 
whether limits of series are themselves spectral lines, on 
the principles of the quantum theory, cannot, at least at this 
juncture, be examined in the light of experiment, and 
that it must remain a matter of deduction from other 
phenomena. 

We find it necessary, as stated, to disagree with the 
hypothesis, explicitly indicated several times by Sommerfeld 
and others, and implicitly assumed at least by the remaining 



Binding of Electrons by .Atoms. 195 

writers on tlie quantum theory of spectra, that the energy W 
is always completely determinate when all the momenta are 
quantized. This can be disproved not only for fictitious laws 
of force in an atom, but for laws which must actually occur 
in systems with an existence, if only a temporary one. 

Consider, for example, a simple doublet and an electron 
in orbital motion about it. Regarding the doublet as 
stationary, and of moment M, its external potential is 

M cos 
~ r* 

when it is situated at the origin, with its axis along the axis 
of z, using spherical polar coordinates. The equation of 
energy for an electron moving in its presence is 

im {r 2 + r 2 fl 2 + r 2 sin 8 6 j> 2 } 4- M * C 8 ° S ° = - W. 

The momenta are, in the usual notation, 

BT . BT %d 

Pi = 57. = mr > Pi = ^ = mr .°> 

Jh — ;— r- — mv 2 sin 2 0$, 
09 

so that 

Me cos 6 



I r r L siir V J r l 



= -w. 



Xow <f> is a speed coordinate as usual, so that 

p z == const. = n-Jifeir 

when subjected to the quantum relation: n x being an integer. 
For the Jacobi solution, we must also take, in separating 
variables, 

P2 2 + ~P\a + 2mMe cos e =/3 
£ suv 6 ^ 

where /3 is constant, and 

>{pi 2 +^} = -w. 



Thus 



-2mW-J. 



02 



196 Dr. J. W. Nicholson on the 

With a positive W, the motion is not real. Thus W must he 
negative and the path necessarily extends to infinity. A 

critical value of r is \J _ w , and the other is infinity. 

The phase-integral for p 1 is 

pi dr 



*J 



v -2mW 



which is infinite, but nevertheless independent of W. For 
writing 

it becomes 



2m W 



= 2/^(1-^ 

A finite integral is secured, — Epstein's procedure, Joy 
instance,— by. using the phase-integral not for p r , but for 
Pi~~ {pi)r=x, which in the same way yields 

again independent of W. Now ft is quantized, or expressed 
definitely in terms of integers already, from the phase- 
integral for the momentum p 2 . The phase integral for pi 
can only, in this case, lead to another expression of similar 
type for j3 t but to no expression for W. It is not at all clear 
that the two expressions for f3, also, can both be valid 
simultaneously. 

This possibility has hitherto apparently been overlooked 
by authors in this subject. 

No case has, however, been noticed in which W is inde- 
terminate for a finite path. One very important conclusion 
is that the whole investigation is valid for a negatively 
charged atom with a distant electron. 

We proceed now to discuss the possible existence of 
definite paths with a positive total energy and infinite 
extent, for a single electron around a nucleus of charge ve, 
situated at the origin. This is Epstein's problem, which he 
treats as only two-dimensional. The energy equation is 



•Z y, 2 



C r r z sin- 6 ) r 



Binding of Electrons by Atoms. 197 

where W is positive, and represents the total energy, and 
the p's are the momenta. 
We have thus 

p. d = const. = -i-, 
Lit 



sin 2 0' 



sin 2 
being clearly positive, 

The phase-integral for p 2 is 

nji = \p, d0 = 2\ d0 . \J /3 2 - 

the limits being the suitable values of for which ^ 2 = 0. 
The factor 2 represents the double journey in this co- 
ordinate, 

sin^=^, 

where ifr is one of the limits, and the other admissible value, 
for a real integral, is 7r — -v/r. Thus 

n 2 h = 2^ d0 V£^3~7sin 2 







Write 






P 2 




w = sm^ &) + — o cos w > 




2?3 


and we have 




n 2 h 


A Q /3 2 — ^ 3 2 f ff/2 COS 2 0) di&> 
— ^P 2 fi>2 ? 

^ 3 -° sin 2 o)+- 2 cos 2 o) 




^3 2 



or with tan co = t, 



n q h = -h; 



ft J. (1+^(1^) 

= 4 J 8(tan- 1 *-§tan- 1 %'l 



198 Dr. J. W. Nicholson on the 

whence 

£ = 27T + p 3 = v 1 ^ 71 *)^' 

these integers being thus additive, in the usual way. 
The phase-integral for p 1 is 



j dr^/2 



n 3 k = \ dr\/ 2mW + 



/3 2 



if we seek to quantize ^ as it stands. The limits would then 
be a positive value of r and infinity, for half the path, and 
the integral would be infinite. But it is clearly necessary 
to suppose that when the electron is at infinity, out of range 
of action of the nucleus, it should not be subject to a quantum 
relation, so that (j»i) r =« is not affected by the rule, and only 
the variable part 

P1-O1X0 

is so affected. Yet this question of quantizing p ± presents 
some difficulties in whatever way it is suggested that it 
should be effected, and we consider that Epstein's discussion 
of the matter is very incomplete and not logically justifiable 
in its mathematical procedure. We shall thus consider 
various alternatives which may give a finite phase-integral. 

Now the actual r-path is not a passage from r = oc (say) to 
r = ao and back, and the phase-integral is not twice the 
defiuite integral between these limits. The electron goes 
from a limiting radius to infinity, and back to the same 
radius elsewhere, and the passage through infinity distin- 
guishes this phase-integral from those which occur in the 
other coordinates. 

We must, of course, also remember that the sign of p x 
depends upon the part of the path concerned, — whether the 
electron is departing or returning. The critical value of r 
is the positive root of 



1 _ mve 2 + \/m 2 v 2 e 4 +2mWj3 2 _ 1 
r ~{P~ ~ ~u 

Writing, generally, with a new variable <f> 

1 _ mve 2 _ AiiW + 2 
r ~W~ ~ V ~gi 



— lilt/ c. I 'V I' 1, v £■ l^ —II' YT kj JL . 

r Jgi = ~ a ( sa ?> 



1 mve* /m 2 v 2 e± + 2mW6 2 

V & ~ cos 9> 



Binding of Electrons hy Atoms. 199 

we have $ = in the critical position (perihelion, in the 
usual terminology), and 

mve 2 1 /m 2 v 2 e 4 + 2mW8 2 
cos< t>=--pr/'\/ ' —p- -=cos77 (say), 

when r=oo. 

What is required for the correct evaluation of the phase- 
integral is a continuous variable which shall change in one 
direction, — and thus give a definite integral, — as r goes to 
infinity and returns, the sign o£ p x being automatically taken 
into account, — or the sign of pi— (piXc when (pi) m is not 
zero as in a parabolic path. The new variable cj> has this 
property, and ranges from zero to 2tt as r goes through its 
changes. We have denoted its value, when r=co, by r) 
above, where rj is evidently an obtuse angle. 

The phase-integral for jt^ alone would be 



r2 V 

n 3 h=\ 



xrr 2mve 2 B" 1 
dr \/ 2mW H S; 



(the square root being properly interpreted in different 
regions) where 

- = ~^r -f- 7^ V m Ve 4 + 2m W/3 2 cos # 
r (3 1 p z 

= ™^ + ~ VmW + 2mW/3 2 cos 77, 

and we find 

dr= T . *? + ** - v , £, p VZmW + mVeS 
(cos 9 — cost;) 4 // 



V 



n TTr 2vme 2 /3 2 q . , 

2m W + ^ = £ sin <£. 

r r p. • 



If the integration were continuous throughout, — as as- 
sumed by Epstein, — we should thus have 



n z h= j3 \ 



sin 2 (f> d(j> 
J (cos<£— cost;) 2 

sin 2 $ d$ 



(cos<£— cosr?) 2 ' 
which is an infinite integral, as would be expected. 



200 Dr. J. W Nicholson on the 

If we merely quantized over the finite part o£ the hyper- 
bola, — another possible suggestion, — we should have 

n s k= fi ( f " + f * ) ,. *}»'* v d4> 

6 I Jo } 2n-n) (cos </> -cost;) 2 r 



=4' 



sin 2 </> dcf) 



(COS <j>— COST;) 5 



which is again infinite. 

The nature of the first infinity merits a remark, however, 
for it is independent of rj and therefore of W. For 



d<f> 

cos?; 



j_C* sin 2 <t>d<l> r sin </> ~| __ C n cos <$> 

J (cosc/>— cos*;) 2 — |_cos<£— cost;J J cos<£ — 

r sin<i "1 C* d<b 

= \ — — — it— cost;! -r-t- . 

|_cos <p — cos 7] J f cos (p— cost; 

The principal value of the last integral is well known to 
be zero, for all values of t;, so that the last term is zero. 
Our equation would be 

^=-2^8+ r f* i, 

Lcoscp— cost; J 

where the principal value of the bracket must be taken, 
i. e. it is to be interpreted as 

Lu { r :»+ r v r f» r }. 

I Lcos — cos t; J LCOS0— COS^Jjj+e J 

This becomes 

^ C e sin t; e sin t; J 2e 

which, though infinite, is an infinity independent of rj and 
therefore of W. We have another aspect of the indeter- 
niinateness of W for such paths. 

Our fundamental objection to Epstein's mode of integration 
may now be introduced. He integrates />i— (pO^? an d not 
pi, but this fact does not affect the question. For as (/> ranges 
between and 27r, if p 1 — /(<£), we have p ] varying con- 
tinuously with 6, and remaining positive, till 6 — r\. Then/?! 
becomes —f(<f>i) when <f> = 27r — ^>i on the return journey 
after <£ = 27r— -t;. Between (/>=?; and <£ = 27r — t;, the value 
of r should be infinite, and p x changes from V^mW to 
— v 7 2wiW, as in the figure. 



Binding of Electrons by Atoms. 201 

The variation of p x between + v/2mW at infinity is the 
source of trouble, and it takes place while - =0. 




p, = - VJmW 

e = 277—77 



Epstein takes twice the integral from </> = to <£=7r, but 
according to the substitution formula, r is negative when <£ 
goes frpm 77 to 77-, and negative values of r are clearly not 
permissible. A suitable integration for the infinite region 
cannot in fact be effected, and any supposition of a suitable 
variable in place of <£>, for the change of p 1 at 00 from 
V2mW to — \Z2mW, would be entirely arbitrary, — but as 
it could not lead to a finite phase-integral, we pursue the 
matter no further. 

These considerations, nevertheless, have considerable force 
when, thrown back as we now are upon the necessity, if the 
quantum theory is applicable, of using p l — (pi) mJ we attempt 
to quantize this. 

We have, when # = 77 

k>i).= ^2mW 
= -* sin 77, 



where g= \/m 2 i>V + 2m W/3 2 as before. 
And when </> = 27r — 77, 

• (#)•= £ sin (27T-77) = - I sin 77. 

From </> = to $=77, 

Pi - (Pi)oo = I ( sin 9 - sin f) • 
From <f) = 27r — n to </> = 27r, 

Pi - (Pi)* = + I ( sin * + sin ^ 
and from (p = r) to $ = 27r — ?7, 



202 Dr. J. W. Nicholson on the 

With the value of dr, the phase-integral 

n 3 h=\ dr( Pl -{p,)J 
Je=o 

breaks into three parts, thus 

Jo . (cos </>- cost;) 2 ^J„ Y 

T 271 " sin (ft (sin </> + sin nMcfr 
J 27r _„ (cos</>- cost;) 2 

Jo (cos</>~ cost;) 2 ^J „ 

by a simple transformation. 

Finally, the only accurate phase-integral is 

f' sin»(»in»-Bi n ^)^ 

,.J (cos 9— cost;)"' 
while Epstein gives, in our notation, 



nji 



— 28 f "" sm< ^ f s i K< fr~ sin?;) 
~ J (cos <j>— cost;) 2 



the part of his range from r/ to it involving a meaningless 
negative value of r, and violating p 1 = (p 1 ) x though the 
moving electron is at infinity. The principal value of 
Epstein's integral is, using the indefinite integral for the 
function in the form, readily obtained by parts, 

J sin $ (sin ff>— sin 7;) 
(cos <£-- cost;) 2 ™ 

sm 0— sin 7} , , , 1 2 

= 7 '- —6+ COt T) . lOge S T 

cos </> — cos t; r n I . rj+(p 

of the type 

^ = 2^1-1^—1—1 
I 7rsmT;J 

or 2 _ (wi + rc 2 + rc 3 ) 

7r sin rj n 1 -\-n 2 ' 

and ultimately 

2 meV J 1 



w= 



^ 7J "" / . . \2 / 

— (Tl! + W 2 + %) - (^1 + W ? / 



— generalized from his value which relates only to a plane 



Binding of Electrons by Atoms. 203 

hyperbola. We have the sum n\ + n 2 of the angular quanta 
in place of his single integer. 

But this formula, with all the applications he makes to 
characteristic 7 radiation, is not tenable, as resting on a 
mathematical error. Its apparent success appeared at one 
time to the writer to justify it as an empirical formula, in 
spite of his independent investigation, outlined above, indi- 
cating the impossibility of quantizing such orbits. Close 
examination, however, of the calculations of 7 radiation and 
so forth made it clear that they' were in several cases 
illusory, and determined more by order of magnitude than 
by the nature of the formula. 

There is one convincing argument against the formula, 
however. It should give an emission spectrum for all values 
of W], n 2 , n s and wi 1: m 2 , m 3 making 

— W(m l9 m 2 , m. d ) +W(n 1 , n 2 , n 5 ) 
positive. This can be tested in great numerical detail on 
the spectrum of a hydrogen atom, and the test fails entirely. 
No spectrum line is found, — in the secondary hydrogen 
spectrum, — in any of the assigned positions. Thus the 
formula really fails as an empirical one. 

We have seen above that it must be replaced by 

J (cose/)- cos 77) 2 






•2/5 



m 6— sin 79 , , ■ , J 

, — - — <P + C0t 77 l0£ e < - 

cos 6 — cos 77 r &e ) . 

- sin 




which is logarithmically infinite. 

The attempt to obtain a finite phase-integral, in this 
manner, in fact fails, and we must give up the hypothesis 
that even the variable part of p 1 can be quantized for the 
infinite path. 

It is not difficult to see that this conclusion is general for 
an} r infinite path which is possible for an electron about a 
physically existent atom, whose nucleus can always be 
regarded, for the present purpose, as a superposition of free 
charges and a set of doublets. We have demonstrated the 
result for a single free charge, and previously for sets of 
doublets. Further analysis of the more general case does not 
seem necessary, and could readily be supplied by the reader. 

Our conclusion must be as follows : — 

A determinate and finite value of W cannot be obtained 
for an electron moving about any atomic nucleus, if the path 
involved takes the electron to infinity. 



L 204 ] 

XIV. Theoretical Aspects of the Neon Spectrum. 
By Laurence St. C. Broughall *. 

T^HE object of this paper is to attempt to explain the 
spectrum o£ neon in a manner somewhat similar to 
that used by Bohr f in his explanation o£ the reason for the 
existence of the Balmer series in the hydrogen spectrum. 

The principle on which this hypothesis rests is that when 
an electron rotates in a fixed orbit it does not radiate energy, 
although the principles of electrodynamics state that it 
should ; if, however, the electron changes from one orbit 
to another, then energy is emitted, provided that the kinetic 
energy of the electron is less in the second orbit than in the 
first. 

In order to account for the spectrum, it is assumed that 
the energy emitted is numerically equal to the product 
of the frequency of the spectral line produced and the 
quantum constant. We thus obtain the equation 

E = nA, 

where E = energy emitted, n = frequency of the resulting 
radiation, and h = quantum constant. 

In the case of hydrogen, it was assumed that the orbit of 
the electron was circular, and then the attractive force 
between nucleus and the electron due to their equal and 
opposite charges was balanced by the centrifugal force of 
the electron due to its rotation about an axis passing through 
the nucleus. 

The energy of the electron can thus be found for any 
radius of orbit. When the electron changes its orbit, it 
moves to one with a radius which is an exact multiple of the 
radius of the original orbit. In this manner the change of 
energy due to a change of orbit can be found, and then, 
using the equation given above, it was shown by Bohr how 
the constant of the Balmer series could be found ; and the 
value so obtained agreed extremely well with that found by 
experiment. 

In the case of neon, we are dealing with an atom which 
contains more than one electron ; and since the atomic 
number is 10, it follows that if the atom is to be neutral, 
then there must be 10 electrons present to annul the excess 
of 10 positive charges in the nucleus. 

* Communicated by the Author. 
t Phil. Mag. vol. xxix. p. 332. 



Theoretical Aspects of the Neon Spectrum. 205 

In order, therefore, to study the atom, it is. essential that 
the electrons be given definite position relative to one 
another. This has been undertaken by Langmuir *, and 
there is considerable evidence in favour of the postulate that 
eight of the electrons arrange themselves at the corners of a 
cube at the centre of which the nucleus is situated. The 
other two electrons are imagined to lie within this cube, 
probably on a line joining the mid-points of any pair of 
opposite sides. If we make use of this hypothesis, and 
further are in possession of data which will allow us to 
find the length of the diagonal of the electron cube, 
then it was shown by the author f that it is possible to 
calculate the angular velocities of the electrons about the 
nucleus. 

Siuce the determination of the spectral lines is an ex- 
tension of the matter given in that paper, it will be advisable 
here to state the principles on which the calculations of the 
electron frequencies depend. 

It has already been stated that the two inner electrons will 
probably lie on a line joining the mid-points of any pair of 
opposite sides. If this be the case, then the electrical forces 
acting on the outer electrons due to the other electrons in 
the outer shell, and to the two inner electrons, will be the 
same whichever electron we take, provided that the two 
inner electrons are equidistant from and on opposite sides of 
the nucleus. 

The next consideration was the axes of revolution. As 
before, it was desired if possible to get the forces acting on 
the outer electrons due to centrifugal action the same for all 
of the electrons. If we take as axes the three lines which 
pass through the mid-points of the three opposite pairs of 
sides of the electron cube respectively, then the above con- 
dition will be satisfied. In the diagram the axes of revolution 
are illustrated by XX', YY', and ZZ'. The inner electrons 
being on the axis XX' will only rotate about two axes. 
It is, of course, quite immaterial which axis the inner 
electrons lie upon. The forces acting on any outer electron 
were then considered, and were taken along the three sides 
of the cube which meet at the point where the electron is 
situated. Now, since the electron must be in equilibrium, 
so the force along each of these lines due to electric;) 1 
attraction and repulsion and also due to the motion in a 
circular orbit must be equal to zero. 

* General Electric Review, 1919. 
t Phil. Majr. Feb. 1922. 



206 Mr. L. St. C. Broughall on Theoretical 

In this manner three equations were obtained, namely 



39e 2 l 



e 2 l 



775 + 



e 2 (r + l) 



e 2 (r — l) 



U 2 T 2s 3 T [(r + /) 2 + 5 2 ] 8 ^ [( r -Z) s + jj s ] 



3/2 



+ m(o 1 2 l + m(o 2 2 l, 



390 s / 



2 £ 2 Z 

= 77* + oTa + 



£ 2 Z 



+ 



4c 3 ~4/ 2 ^2.9 3 ^ [(r + Z) 2 + s 2 ] 3/2 ^ [( r -Z)* + iS]3/* 

+ mo^ 2 / + mcofl, . . . 
J Z e 2 Z e 2 l 



Me 2 l 



+ 



±c z ~4Z 2 " r 2.5 3 ^ [(r + 2 + 5 2 ] 3 / 2 " r [(r-l') 2 +s 2 ] 

+ 171(0% I + 11l(0 2 l. . . 

Fig. 1. 
Y 





l£ 


- 2l 




— »e Q 




c 5 








i\ 


\ 


\ e^ 


~ ~~~~2 


r=^ 


-~-~~J____\. 








i = ^ r 


' c i 










i 
i 








l\^io 




i 










E 


i 










\ e 1 

>?9. J 

l\ 




*4 


e 3 
















e 2 



-z ! 



X 1 



(I.) 
(II.) 
(III.) 



Y 1 

Another equation can be obtained by considering the 
forces acting on either of the inner electrons along the line 
joining the two inner electrons. Equating the forces to 
zero, we found that 



10e s 



4<? 2 (r + I) 



» + 



±e 2 (r-l) 



r 2 ' [ (r + I) 2 + s 2 ] 3 / 2 T [ (r - /) 2 + s 2 1 *' 2 + 4r 2 

-f- m(Oi 2 r + m(o 2 2 r, . . (IV.) 
In the above equations, e = charge on an electron ; 



Aspects of the Is eon Spectrum, 207 

r = radius of orbit described by the two inner electrons ; 
coi = angular velocity about YY' ; co 2 = angular velocity 
about ZZ' ; &> 3 = angular velocity about XX'. I = I length 
side of electron cube ; s = ^ surface diagonal of the electron 
cube; c = ^ diagonal of cube, 'a' and i V are of course 
functions of ' c,' and if the latter is known, then ' 5 ' and 
' I ' can be found, m — mass of an electron when its velocity 
is small compared with that of light. * 

From these equations it was shown by the author that 
an equation involving only ' r,' ' c/ ' /,' and ' s ' could be 
found. In the paper mentioned, '/' and ' s' were not 
expressed as functions of ' c,' but expressing them as such, 

since 1 = — — and s= — ~— , we obtain the equation 
o o 

4-34 r-h'Dllc f, 2-308c-) 



r + 'Dile r . Z'dVSc 1 

" L0'+'577c) 2 + -667c 2 ;p ) r J 

r-'577c f 2'308c^ 5^ 

[(r--577(') 2 + -667c 2 ] 3/2 1 r J + r 



. (V.) 



In a recent article it was shown by Prof. W. L. Bragg * 
that the diameter of the neon atom could be found by an 
inspection of the diameters of the atoms of elements whose 
atomic numbers were near that of neon. It is impossible to 
measure the radius of the neon atom directly, since it forms 
no chemical compounds. The value obtained was very much 
smaller than that found by gas measurements, and the former 
is considered by Bragg to be the distance between the elec- 
trons in the atom — that is to say, is equal to ' 2c/ The value 
obtained by Chapman f from gas measurements is, however, 
the diameter presented by the molecule when in collision 
with other molecules. The difference is due to the fact 
that in molecular collisions in the gaseous state the outer 
electrons of the molecules do not come into contact}:. 
Using Bragg's value we have 2c = l'30 X 10~ 9 cm., and on 
substituting in equation (V.) we have a means of obtaining 
the fundamental value of l r.' 

An inspection of equations (II.) and (III.) shows at once 
that o) 1 = o) 2 and equations may be obtained for co 1 and co 3 . 

* Phil. Mag. vol. xl. p. 169. 

t Trans. Roy. Soc. A. vol. 216, p 279. 

X Rankine, Proc. Roy. Soc. vol. xcviii. p. 360. 



208 Mr. L. St. 0. Broughall on Theoretical 

These equations take the form 

e 2 ( 39 , r r + l r — l ~]\ — 2 

2mr 1 4r 2 L{F+F+^P + ■{ (r " 0' + s ' P /2 J J ~~ ^ ' 

.... (VI.) 
e 2 r f 1 11 

ml { [( r -Z) 2 + 5 2 ] 3 / 2 " [(r + /) 2 + 5 2 ] 3 / 2 J = G, i 2 - ft) 3 2 . (VII.) 

Using a slightly different value for ' m ' from that used in 
the previous paper, we obtain the following values : — 

6*19 x 10" 9 cm. 6-034 x 10 16 rad/sec. 4*290 x 10 16 rad/sec, 

'm' being equal to 9*005 x 10 _28 grm. and £ = 4'774 x 10 -10 
E.S.U. 

These figures refer to the neon atom when in its normal 
state. There is some doubt as to whether they apply without 
modification in the gaseous state, but certain assumptions 
are made later in this paper which leads one to the conclusion 
that if the atom is larger under natural conditions, then the 
only result will be the elimination of certain spectral lines 
in the ultra-violet. When the atoms of the neighbouring- 
elements were submitted to measurement, they constituted 
a solid body ; it is, therefore, quite conceivable that modi- 
fication will occur if the element becomes gaseous. 

In order to explain the nature of the spectral lines, we 
have to consider the change of energy due to a change of 
orbit, energy being emitted when the orbit increases in 
diameter. 

Bohr, as already stated, imagined in the case of hydrogen 
that the radius of the orbit increased by constant multiples 
of the radius of the initial orbit. To adopt such a plan in 
the case of neon would lead to the emission of spectral lines 
of a frequency which would only give ultra-violet lines under 
reasonable circumstances. Further, there is no reason why 
the increment should be of such a nature, and the hypothesis 
used in our case is that the spherical shell formed by the 
inner electrons increases in radius until the shell has a 
radius equal to that of the initial outer shell of electrons. 
In order that equilibrium may remain, it is essential that 
the outer shell also expands to an extent which can be 
calculated from equation (V.). 

The initial increment is of the nature of 3xl0~ 10 cm. 
This process of expansion continues again and again, the 
inner electrons always occupying the orbit previously 
occupied by the outer electrons. 




Aspects of the Neon Spectrum. 



209 



Having thus found a definite value for l r' corresponding 

to a definite value for * c' we are now in a position to 

calculate the values of the angular velocities about the 

... 
several axes. To do this, equation (VI.) is first applied ; 

and having found ' g>jV the value obtained is substituted in 

equation (VII.), and &) 3 2 obtained. In calculating the value 

of ' c,' use is made of the fact that the ratio c : r is practically 

constant ; such ratio values are used in Table II. The 

closeness of the ratio figures and the absolute figures is 

shown in the appended table. 

Table I. 



6-50xl0" 9 cm. 
12-33 „ 
1501 



c (ratio method). c (from eqn. V.). 

6-83xl0 _9 cm. 6-83xl0~ 9 cm. 

12-95 „ 1295 

15-77 „ 15-77 

It will be seen that the agreement of the figures is so 
nearly exact as to warrant their use, remembering that our 
fundamental value of { c ; has not been obtained experi- 
mentally. 

Table II. shows in columns II. and III. the values of 
l c' and t r/ and in columns IV. and V. the values of w^ 
and ft) 3 - respectively. 

Table II. 



I 


II. 


III. 


IV. 


1 


No. 


Eadius 
Outer Orbit = c. 


Radius 
Inner Orbit = r. 


o h \ 


, 1 


1 ... 


6-50xl0 _9 cm. 


6-19xl0" 9 cm. 


36-41 X10 32 


18-40 xlO 32 | 


2 ... 


6-83 


6-50 


3141 „ 


15-84 


3 ... 


717 


6-83 


27-08 „ 


13-65 „ 


4 ... 


7*54 „ 


717 


2336 „ 


11-78 „ 


5 ... 


7-92 


7-54 


20-15 „ 


10-16 „ 


6 ... 


8-32 


7 92 


17-38 „ 


8-768 „ 


7 ... 


8-74 


832 


15-00 „ 


7-568 „ 


8 ... 


9-18 


8-74 


12-93 „ 


6-528 „ 


9 ... 


964 


9-18 


' 1115 „ 


5-620 „ 


10 ... 


1013 


964 


9-625 „ 


4860 „ 


11 ... 


10-64 


10-13 


8-299 „ 


4182 „ 


12 ... 


11-18 


1064 


7160 „ 


3613 „ 


13 ... 


11-74 


1118 


6170 „ 


3-109 „ 


14 .. 


1233 


1174 


5-326 „ 


2-684 „ 


15 ... 


12-95 


12-33 


4-602 „ 


2-322 „ 


16 ... 


13-60 


12-95 


. 3-970 „ 


2-002 „ 


17 ... 


14-29 


1360 


3428 „ 


1-729 „ 


18 ... 


15 01 


14-29 


2-954 „ 


1-489 ,, 


19 ... 


15-77 


1501 


2-550 „ 


1-286 „ 


20 ... 


16-57 


1577 


2-201 „ 


1113 „ 


21 - 


17-41 


1657 


1-897 „ 


•9581 „ 

1 



Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. 



210 Mr. L. St. C. Broughall on Theoretical 

Let us now consider the energy of an electron in the 
outer shell when the diameter of the shell = 2^. 

Let the angular velocities about the axes YY' and ZZ' 

=w 1 . 

Let the angular velocity about axis XX / = W 3 . 

Since the diameter of the shell is equal to ' 2c 1? ' it follows 
that the radius of the electron orbit =Sj where Sx = C! v 6/3. 

Using the above notation and remembering farther that 
the kinetic energy of a particle describing a circular path 
of radius ' R ' with an angular velocity 'W is equal to 
^MR 2 W 2 , where M is the mass of the particle, we find that 
the kinetic energy due to rotation about the axis XX' 



= ±mS{ 2 W 



and the kinetic energy due to rotation about the axis YY' 
plus that due to rotation about ZZ' 

= mS 1 2 W 1 2 since Wi = W 2 . 

Therefore the total kinetic energy of the particle is 
equal to 

>S 1 2 {2W 1 2 + W 3 2 }. 

In the case of an electron in the inner shell where the 
radius of the orbit =Rj, we have the kinetic energy of 
the electron due to its rotation about the axes YY' and 
ZZ'=E X where 

E 1 =wR 1 2 W 1 2 . 

Now let the inner shell expand until it occupies the space 
previously occupied by the outer shell — that is to say, until 
Ri = Ci, then kinetic energy in new orbit = E 2 » 

Where E 2 = ?nC 1 2 W 11 2 , 'W n ' being the new angular 
velocity about the axes YY' and ZZ', the change of energy, 
= E 1 -E» 

= m(R 1 2 W 1 2 -C 1 2 W 11 2 ). 

Meanwhile the outer electrons have moved further away 
from the nucleus, and now the outer shell has a radius = G 2 , 
and the orbit of the electrons is now S 2 . 

Therefore the energy in the new position is equal to 

*mS 1 *[2W 11 J + W 1 ,»]. 
Where ' Wi 3 ' is the new angular velocity about the axis 



Aspects of the Neon Spectrum. 21.1 

XX', the change of energy due to change of orbit is 
therefore equal to 

im{S l 2 [2\V + W 3 2 ]-S 2 2 [2W„ 2 + W I3 2 ]}. 
Now, by Bohr's assumptions we have the equation 
Energy Emitted = Frequency x h. 

Now, the frequency of a light-wave =c/X where l c' is 

the velocity of light and i \ i is the wave-length. 

Therefore 

ch 
Energy Emitted = e= — , 

ch 
or X= — . 

e 

Now, in the case of an inner electron 

ch 



\= 



mCR^WV-L^VVn*)' 



giving a series of lines for different values of R. 
In the case of the outer electrons, 

2ch 

X ~ m{S{\2 W * + W 3 *) + S 9 »(2 W n 2 + W„«) } ' 

giving a second series of spectral lines. 

Table III. shows the energies corresponding to definite 
radii. Column II. shows the energy content of an inner 
electron on the left, and that of an outer electron on the 
right. Column III. shows the change of energy, and 
column IV. shows the wave-lengths of the spectral lines 
produced. It should be stated here that the energy under 
consideration is the energy of one electron and not of I he 
whole shell. It has been stated that there is a possibility 
oE the atom not being in its normal condition to begin with, 
owing to its gaseous condition. If, however, it has expanded, 
then instead of starting with an atom whose diameter is 
1'oOxlO -8 cm., we start with one whose radius is in all 
probability equal to one of the radii given in Table II. 
If this is so, then the only change produced will con- 
sist of the elimination of some of the lines of higher 
frequency. 



P 2 



212 



Mr. L. St. C. Brouffhall on Theoretical 



Table III. 



I. 


II a. 


It b, 


III a. Ill b. 


IV a. IV b. 


No. 


Energy per electron. 


Energy Difference 

(ergs). 


Wave-length of 
Spectral Line. 


1 .. 


Inner. 


Outer. 


Inner. 


Outer. 


Inner. 


Outer. 


1-255X10" 10 


1-157 XlO- 10 










2 .. 


1-195 „ 


1-101 „ 


6-0xl0- ]2 


5-6X10" 12 


3272 A 


3505 A° 


3 .. 


1-137 „ 


1-048 „ 


5-8 „ 


5-3 „ 


3386 „ 


3704 ;, 


4 .. 


1-082 „ 


•9970 „ 


5-5 „ 


5-1 „ 


3570 „ 


3850 ,/ 


5 .. 


1030 „ 


•9491 „ 


5'2 „ 


4-79 „ 


3776 „ 


4100 „ 


6 .. 


•9801 „ 


•9035 ,; 


4-96 „ 


4-56 „ 


3959 „ 


4305 „ 


7 .. 


•9337 „ 


•8604 „ 


4-67 „ 


4-31 „ 


4205 „ 


4555 „ 


8 .. 


•8886 „ 


•8187 „ 


4-51 ., 


4-17 „ 


4353 „ 


4709 „ 


9 .. 


•8457 „ 


•7793 „ 


4-29 „ 


3-94 „ 


4576 „ 


4983 „ 


10 .. 


•8055 „ 


•7423 ., 


4-02 „ 


3-70 „ 


4884 „ 


5307 ,.. 


11 .. 


•7665 „ 


'7063 „ 


390 „ 


3-60 „ 


5034 „ 


5454 „ 


12 .. 


•7297 „ 


•6723 „ 


3-68 „ 


3-40 „ 


5335 „ 


• 5775 ,. 


13 .. 


•6940 „ 


•6396 „ 


3-57 „ 


3-27 „ 


5499 „ 


6005 „ 


14 .. 


•6609 „ 


•6089 „ 


3-31 „ 3-07 „ 


5932 „ 


6396 ., 


15 .. 


•6298 „ 


•5803 „ 


311 „ 


2-86 „ 


6313 „ 


6864 „ 


16 .. 


•5995 „ 


•5525 „ 


303 „ 


2-78 „ 


64S0 „ 


7063 „ 


17 .. 


•5709 „ 


•5260 „ 


2-86 „ 


265 „ 


6864 „ 


7409 „ 


18 .. 


•5430 „ 


•5003 „ 


2-79 „ 


2-57 „ 


7037 „ 


7640 „ 


19 .. 


•5172 „ 


•4766 ,, 


2-58 „ 


2-37 „ 


7610 „ 


8285 ,. 


20 .. 


•4927 ., 


•4540 „ 


2-45 „ 


226 „ 


8013 „ 


8688 „ 


21 .. 


•4689 „ 


•4321 „ 


2-38 „ 


2-19 „ 


8249 „ 


8966 „ 



Owing to the complexity of the neon spectrum, it would 
be useless to attempt to compare our calculated lines with 
those found by experiment ; indeed, it would be deleterious 
to attempt such a comparison, since the impression would be 
given that there is a definite line in the spectrum which 
corresponds to one of our calculated lines. Emphasis may 
only be laid upon the fact that our series produce lines 
in the visible part of the spectrum, which do not compare 
unfavourably with those obtained by experiment. Reference 
to Table III. will show at once that only forty lines have 
been determined between X = 3272 A and \ = 8966A, 



L 



Aspects of the JSfeon Spectrum. 213 

whereas there are many more lines in existence. These lines 
can only be explained by the fact that when the spectrum is 
obtained, large numbers of ionized atoms exist, and under 
such circumstances our fundamental equations no longer 
hold. 



The discussion of the properties of ionized atoms is very 
complex, since the possible degrees and modes of ionization 
are very numerous. The first case which comes under con- 
sideration is the atom which has lost one electron, thus 
leaving an excess of one positive charge. 

It is very probable that one of the outer electrons will be 
removed, thus leaving seven electrons in the outer shell. 
Now, it seems probable that the angular momentums of the 
remaining electrons will suffer no change, the light pro- 
duced during ionization being due solely to the change of 
energy of the electron suffering removal. We are thus left 
with seven electrons, each possessing the same angular 
velocity. It is a matter of considerable difficulty to arrange 
these electrons, and it is impossible to arrange them on a 
spherical surface without the force acting on an electron 
varying with the electron taken. We are therefore obliged 
to separate them into different shells. 

Now, since the angular momentums of our seven outer 
electrons are the same, it follows that, if they are not on the 
the same spherical surface, then they must be in motion 
relative to one another. Under such conditions the positions 
of the electrons will vary with time. 

Owing to the complexity of such a case, it seems impossible 
to treat the case mathematically without more experimental 
evidence. There are further atoms present which have been 
ionized to a greater extent, thus losing several electrons. 
Similar difficulties are met with in the cases of atoms with 
five or six electrons in the outer shell as in the case of seven 
electrons. The cases in which four or six electrons have 
been removed are, however, considerably simpler, since the 
electrons may then be given positions on a spherical surface 
such that the force acting on an electron is not dependent 
on the electron taken. In general, the atoms ionized to so 
great an extent will be comparatively few. In all our cases 
of ionized atoms, it must be remembered that it is not only 
the normal atom that is ionized ; an atom may have given 
out several spectral lines before it becomes ionized. So a 






214 Theoretical Aspects of the Neon Spectrum. 

large number of lines will be obtained depending upon the 
state of the atom when one or more electrons are removed 
from it. There is another form of ionization which is worthy 
of consideration. That is the case in which an electron has 
succeeded in penetrating the atom and reached the nucleus, 
thus temporarily reducing the positive charge and therefore 
giving a negative ion. 

The fundamental mathematical expressions for such a case 
are found by extending our formulae for the neutral atom 
for the case where the charge on the nucleus is ' n ' instead 
of ten. 

The angular velocities about the several axes will remain 
unchanged ; and so only two equations will be required to 
determine the new values assumed by ' c ' and ' r,' the radii 
of the outer and inner shells respectively. 

We have three available equations ; and expressing them 
in the notation previously used, we obtain : 

(4w-l> 2 Z e 2 eH_ e 2 (r + l) e 2 (r~l) 

4c 3 " ~ U 2 + 2s* + [ (r +■ l) 2 + s 2 ] *' 2 ~~ [ (r - I) 2 + s 2 ] 3 / 2 

+ 2mo?l, .... (la.) 

(4n-l> 2 / e 2 eH e 2 l eH 

4c 3 ~ - 4/ 2 + 2? + [(r + iy + s 2 y/ 2+ [{r-l) 2 + s 2 ] s < 2 

-hmco 1 2 l-\-?nco :i 2 l, . . (II a.) 



ne* <ke 2 (r + l) 4<? 2 (r-Q 

+ Vt*. 7\2 i .213/2 + 



+ 2mft> 1 V, . . . .(Ilia.) 

which are obtained from equations (I.), (II.), and (IV.), 
replacing the nucleic charge of ' 10<? ' by ' ne 9 and remem- 
bering that &) 1 = G) 2 . 

The result of the alteration will be that the electrons will 
move further out from the nucleus, since i n } is of necessity 
less than ten. In consequence of this, the frequency of the 
spectral lines produced by such ionized atoms will be of a 
lower frequency than those produced by the neutral atom. 
There will in consequence be a larger number of lines in the 
part of the spectrum of greater wave-length. It is for this 
reason that there are so many lines in the orange, and red 
in the case of neon. 

Feb. 13, 1922. 



1. 






[ 215 ] 

XV. Absorption of .Hydrogen by Elements in the Electric 
Discharge Tube. By F. H. Newman, Ph.D., F. Inst. P., 
A.P.C.S., Head of the Physics Department, University 
College, Exeter *. 

1. Introduction. 

rjpHE phenomenon of the disappearance of gas in the 
JL electric discharge-tube, and in the presence o£ incan- 
descent filaments, has received much attention recently owing 
to its importance in technical applications. Langmuir f has 
shown that hydrogen disappears from a vacuum tube in 
which a tungsten filament is heated above 1000° C. # This 
fact has been utilized by him in the removal of the last traces 
of gas in valves, and the effect has been termed a " cleaning 
up " one. The pressures at which he w r orked were very low ; 
for example, he found that the pressure in a tube was lowered 
to 0*00002 mm. of Hg. Other gases, including nitrogen and 
carbon monoxide, are removed in a similar manner, and 
molybdenum, when incandescent, has the same effect as 
tungsten. In all cases Langmuir found that the cooling of 
part of the apparatus by means of an enclosure at liquid- 
air temperature greatly accelerated the rate of disappearance 
of the gases. In addition he noted an electro-chemical 
" clean up," which occurred at much lower temperatures of 
the filament, when potentials of over 40 volts were used 
in a way that caused a perceptible discharge through the 
gas. 

More recently Campbell, conducting work for the General 
Electric Company J and using incandescent filament cathodes 
in electric discharge-tubes, has made an exhaustive study of 
the " clean up " effect, and has come to the conclusion that 
there is much evidence for believing there exists an electrical 
action which is quite independent of the thermal action, and, 
providing the temperature of the filament is kept below that 
at which the chemical " clean up " occurs, the effect appears 
to be one dependent only on the electrical discharge. In the 
case of the disappearance of carbon monoxide there is proof 
of the conversion of this gas into carbon dioxide, and the 
action takes place more rapidly when part of the apparatus 
is cooled to liquid-air temperature. This has the effect of 
removing the carbon dioxide by condensation as quickly 

* Communicated by the Author. 

t Am. Chem. Soc. Journ. vol. xxxvii. (1915). 

X Phil. Mag. vol. xl. (1920) ; vol. xli. (1921) & vol. xlii. (1921). 



216 Dr. F. H. Newman on Absorption of Hydrogen 

as it is formed. The presence of phosphorus vapoui 
accelerates the rate of disappearance of all gases except 
the inert ones, and much lower final pressures are attained. 
This, the author believes, is due to the deposition of the gas 
on the walls of the vessel, this deposit then being covered 
with a layer of red phosphorus formed by the electric 
discharge passing through the phosphorus vapour. The 
covering of red phosphorus prevents liberation of the 
hydrogen by bombardment of the ions, and at the same 
time provides a new surface on which further gas can be 
deposited. 

The problem of the disappearance of the gas is a very 
complicated one, owing to the many factors to be considered. 
The walls of the vessel and the electrodes will certainly receive 
some of the gas, although the latter may not disappear in its 
original state. There will be chemical changes occurring in 
the volume of the gas, such as the conversion of carbon 
monoxide into carbon dioxide, and, in addition, any other 
elements present in the discharge-tube, either in the form 
of vapour or solid, will affect materially the rate of dis- 
appearance of the gas and the final pressure reached. 

The author * has shown previously that various substances 
present on the electrodes of a discharge-tube alter considerably 
the amount of gas that can be caused to disappear when an 
electric discharge is passing. In particular, phosphorus, 
sulphur, and iodine cause both hydrogen and nitrogen to be 
absorbed at a very great rate, and a high vacuum is quickly 
produced as a result. This action of phosphorus has been 
used for many years to obtain and maintain very low 
pressures in valves. These three elements stand out as being- 
very effective even at high pressures, but other substances 
which were tested in a similar manner did not appear to 
absorb hydrogen. On the contrary, gas appeared to be 
liberated. This effect can be explained as follows. At 
pressures above 1 mm. of Hg. a certain amount of the gas 
in a discharge-tube becomes occluded within the walls. 
This gas will be liberated when the walls are bombarded 
by the ions produced by an electric discharge. This effect 
will mask any disappearance. If, however, the tube is 
heated almost to the softening point of glass and highly 
exhausted, then on admitting hydrogen at a small pressure 
such as O'l mm. of Hg., very little occlusion of the gas 
within the walls will take place, and on passing the electric 

* Newman, Proc. Hoy. Soc, A. vol. xc. (1914); Proc. Ph. vs. Soc. 
vol. xxxii. (1920) & vol. xxxiii. (1921). 



by Elements in the Electric Discharge- Tube. 217 

discharge practically no hydrogen will be liberated from the 
walls by bombardment with the ions. If there is any 
absorption of the gas, this effect will not be masked by the 
liberation of the gas from the walls or electrodes. 

The object of the present work was the study of the 
behaviour of hydrogen in the presence of various elements in 
a discharge-tube when a current was passing through it. 
The pressures of the gas in these experiments were much 
lower than those used by the author in the papers quoted 
above, but they were much greater than those used by the 
previous investigators — Langmuir and the General Electric 
Campany. 

2. Description of Apparatus. 

At gas-pressures below O'l mni.Hg.it is difficult to obtain 
a current through a discharge-tube unless very high potentials 
are used. A valve also must be placed in the circuit to make 
the discharge unidirectional. This entails further diminution 

Fiff. 1. 



M c LEOD GAUGE 




3=fcdl^b- 



of the current. By using a Wehnelt cathode the potential 
required was greatly reduced. The apparatus employed is 
shown in fig. 1. The incandescent filament was a strip of 
platinum foil 5 mm. long and 3 mm. wide. As the 
discharge-tube had to be thoroughly cleaned after each 
experiment, the cathode was sealed in a glass stopper which 
could he removed when the tube was cleaned. This 
necessitated the use of tap-grease, but the vapour arising from 
it did not appear to afreet the results at the pressures used. 



218 Dr. F. H. Newman on Absorption of Hydrogen 

Previous experiments had shown that elements such as 
sodium and potassium are only effective in causing the 
disappearance of gas in the electric discharge-tube if the 
surface of the element is clean, and if it has been prepared 
in vacuum. Accordingly the substance under test was 
placed on the platinum foil forming the cathode, and after 
the tube had been heated almost to the softening point of: 
glass and exhausted, the element on the foil was vaporized 
by passing an electric current through the latter. In this 
way the substance was then deposited on the inner surface of 
the anode D and an uncontaminated surface obtained. The 
anode was of aluminium and was cylindrical in shape, fitting 
very closely to the glass walls. Enclosing the cathode in 
this way, the effect of the surface of the glass on the 
absorption of the gas was minimized. A side tube B was 
used to contain the easily volatile elements such as phosphorus, 
sulphur, and iodine. An aperture was made in the anode 
opposite the mouth of B so that the vapour ot the substance 
from B could pass through and be deposited on the inner 
surface of D. The pressures of the gas were measured with 
a McLeod gauge. The hydrogen was prepared by the 
electrolysis of barium hydroxide and stored in a reservoir. 
This method of preparing the gas ensues great purity. Any 
oxygen present was removed by passing the gas through a 
bulb containing sodium-potassium alloy. Phosphorus pent- 
oxide in F removed any water-vapour, and of course the alloy 
was effective in this respect also. The gas could be admitted to 
G, which was a known volume (0*051 c.c.) enclosed between 
two taps. A definite volume of gas at a known pressure 
could thus be admitted to the discharge-tube. From obser- 
vations of the pressure in the tube before and after a discharge 
had passed, the actual volume of gas — at atmospheric 
pressure — which had disappeared could be calculated. 

The current through the discharge-tube was kept constant 
by altering the filament current, and was measured with a 
galvanometer. In previous experiments the quantity of 
electricity passing through the tube while absorption was 
taking place had been measured with a water voltameter, 
but in the present work this method was not sensitive 
enough. 

After deposition of the substance on the anode D, the 
tube was again highly exhausted to remove any gases 
liberated from the volatized substance. The tube was placed 
in an enclosure maintained at —40° C. while absorption 
of gas was in progress. 



by Elements in the Electric Discharge- Tube. 219 

Observations were then taken of the changes in pressure 
due to the disappearance o£ the hydrogen when an electric 
discharge passed through the gas. The results obtained are 
shown in the accompanying table. 

The accelerating potential was 94 volts, obtained by using 
small accumulators. The current through the tube was kept 
constant, and was 546 micro-amps. 

Each set of readings corresponds to an electric discharge 
for ten minutes, except in the cases of sulphur, phosphorus, 
and iodine, where the observations were taken at intervals 
of two minutes — i. <?., with sodium the pressure changed 
from 743 xlO" 3 mm. of Hg. to 336 x 10" 3 mm. of Hg. in 
ten minutes, while with sulphur the pressure was lowered 
from 740 xlO" 3 mm. of Hg. to 329 x lO" 3 mm. of Hg. in 
two minutes. 

The amount of hydrogen which would be liberated from 
a water voltameter in 10 mins. by the same current is 
39 x 10~ 3 c.c. at atmospheric pressure. 

As the gas may disappear into the walls of the anode 
even in the absence of any substance on the anode, and 
as the glowing filament may affect the rate of disappearance, 
preliminary observations were always made when an electric 
discharge passed through the tube without the substance 
present on the anode. The volume of gas which disappeared 
owing to these two effects was always very small compared 
with that which was absorbed when the element under test 
was on the anode. 

3. Experimental Results. 

After each element had been tested, the tube was heated 
to 300° C, and the volume of gas reliberated was calculated 
from the observed change of pressure. The amount thus 
recovered varied considerably, but was always less than that 
which had disappeared. This evolved gas was again absorbed 
when a discharge was passed, and it is evidently hydrogen 
in the same condition as it was before disappearance. 

If, after the gas had disappeared, a fresh amount of 
hydrogen was admitted, the volume which disappeared on 
discharge was reduced. For example, with sodium and the 
gas pressure at 743 X 10" 3 mm. of Hg., the vacuum was 
reduced to 96xl0" 3 mm. Hg. before the action ceased. 
Admitting a further supply of gas to the tube, the pressure 
fell from 743 xlO" 3 mm. Hg. to 233 xlO" 3 mm. Hg. 
Repeating the process again, the pressure fell from 
743 xlO" 3 mm. Hg. to 436 x 10" 3 mm. Hg., and then the 



220 Dr. F. H. Newman on Absorption of Hydrogen 



Element. 



Initial Gas 

Pressure. 

mm. H?. X 10-3 



Final Gas 

Pressure. 

mm.Hg.XlO-3. 



Volume of Gas 

absorbed. 

c.c. x 10-3. 



Pressure of 

Gas at which 

action ceased. 

mm.Hg.XlO -3 . 



Sodium 



Potassium 



Sodium-Potassium 
Alloy. 



T743 
1336 

726 
349 

738 
392 



336 
123 

349 
163 

392 

188 



329 
145 

58 



f740 
Sulphur i 329 

f748 

Phosphorus -{352 

1200 

t j- ' 763 

Iodme 1338 

Arsenic {394 

Cadmium j ^o 

Calcium |^ 04 

»»■': {23 

ry J 753 

Zmc J431 

Thallium {560 

j , j\ Hydrogen was liberated and not absorbed 



352 

147 

44 

338 
150 

394 

206 

463 
321 

404 



386 
264 

431 

306 

560 
399 



28 
14 

26 
13 

23 
15 

28 

13 

6 

27 
14 

7 

28 
13 

24 
13 

19 

10 

23 

7 

25 

8 

22 

8 

12 
11 



96 

84 

110 
26 

14 

124 

108 
284 
152 
131 
276 
297 



absorption ceased. There appears to be a fatigue effect 
whereby the actual amount of gas which can be absorbed by 
any surface is limited. This fatigue effect may be due to 
three causes. If the disappearance of the gas depends on 
chemical action, the latter will occur mainly at the surface 
of the element. The formation of a chemical compound will 
thus protect the rest of the substance from the action, and 
the process will gradually cease. If, on the other hand, 
the effect is due to a deposition of the gaseous atoms on 



by Elements in the Electric Discharge- Tube. 221 

the surface, as Langmuir suggests, these atoms will diffuse 
slowly into the substance. The atoms arriving later will 
have less area <>n which deposition can take place. A limit 
to the action will be reached when the number of atoms 
deposited is equal to the number set free by the bombard- 
ment of the surface by the ions. 

After absorption, the proportion of the hydrogen re- 
liberated when the tube was heated to 300° 0. varied 
considerably in different cases, not only with different 
elements, bat also with the same element. This is to be 
expected when it is remembered that the thickness of the 
substance deposited on the anode varied with different 
substances. 

The accelerating potential affected to some extent the rate 
of disappearance of the gas and also the final pressure 
attained. Owing to liberation of the gas by the bombardment 
with the ions, the final pressure reached must depend on this 
reverse action, and the greater the accumulation of the gas on 
the surface of the anode, the greater will be the amount of 
gas evolved. 

With sodium on the anode a potential of 94 volts reduced 
the gas-pressure from 743 X 10~ 3 mm. Hg. to 123 x 10 _3 mm. 
Hg. in the course of 20 minutes. When the potential was 
lowered to 54 volts, the pressure fell from 743 x 10 ~ 3 mm. 
Hg. to 476 x 10" 3 mm. Hg. in the same time-interval. 
The final pressures reached before absorption ceased were 
96xl0" 3 mm. Hg. and 202 xlO" 3 mm. Hg. respectively. 
The current through the discharge-tube was kept constant 
throughout. 

The principles of the disappearance of the gas will be 
discussed later, but there are certain features of the 
phenomenon which can be traced to chemical actions. 

Many of the elements tested combine with hydrogen at 
high temperatures to form chemical compounds which are 
very stable. Any chemical action occurring in the present 
experiments cannot be due to the heat, as the discharge-tube 
was maintained at —40° C, and the incandescent filament 
was always at a lower temperature than that at which 
Langmuir found chemical action occurred with hydrogen. 
The effect may be caused by " activation " of the gas, the 
latter assuming some modification under the action of the 
electric discharge. In the above experiments the amounts of 
gas absorbed were so small that it would be extremely 
difficult to detect the existence of any chemical compound in 
the tube. In order to increase the amount of gas absorbed, 



222 Dr. F. H. Newman on Absorption of Hydrogen 

and test for any chemical compounds formed, a modified form 
of the apparatus was used, as shown in fig. 2. 

Pure hydrogen could b.j admitted to the discharge-tube A 
in small volumes by manipulation of the taps 1\ and T 2 . 
Two strips of platinum foil, about 10 cms. long, were sealed 
in the tube E. These strips fitted closely to the glass 
surface. A potential difference of 600 volts was applied 
between these strips by means of small accumulators. In 
this way the ions actually present in E were removed while 
the discharge was proceeding in A. The tube E communi- 
cated with a mercury cut-off K, and a U tube immersed in an 
enclosure maintained at —40° C. Sodium-potassium alloy 
was prepared in D, and after the whole of the apparatus had 
been evacuated, the alloy was run into C. In this way a 
bright and clean* surface was obtained on the alloy in C. A 




n \ HYDROGEN 



small volume of hydrogen was then admitted to A, the 
mercury cut-off preventing the gas from entering C and D. 
While the electric discharge was passing in A, the hydrogen 
was allowed to enter C by manipulation of the cut-off. 
Admitting successive volumes of hydrogen into A in this 
way, and each time allowing communication with C while 
the discharge was proceeding, an increasing amount of active 
gas entered C, and an effect was observed on the surface of 
the alloy. At first it appeared to be covered with a thin 
white crystalline compound when observed through a 
microscope. This white layer slowly changed, on the 
admission of more active gas, to a dark grey-coloured 
deposit. To show that this surface effect was not due to 
impurities in the hydrogen, previous experiments were 
made, the gas being admitted to C without the electric 
discharge proceeding. There were no surface effects then, 



by Elements in the Electric Discharge- lube'. 223 

so it was concluded that some of the hydrogen assumes an 
active modification under the action of the electric discharge, 
and in this form it is able to form chemical compounds with 
the sodium and the potassium present in the alloy. The U 
tube in the enclosure at —40° C. excluded the possibility of 
the action being due to the heat from the discharge-tube. 
The white crystalline compound which first appears is a 
mixture of the hydrides of sodium and potassium. The exact 
nature of the greyish-coloured product formed afterwards is 
unknown, but it is probably a solution of the hydrides in the 
alloy. 

Water is evolved by an electric discharge when passed 
through any vacuum vessel. It comes from the glass, and 
would not be kept back by the trap cooled to —40° C; for 
at that temperature water substance has a vapour-pressure of 
about 0*1 mm. Hg. The presence of water-vapour " fouls " 
the surface of the alloy, but this fouling gives a black 
deposit on the surface which is quite different from the 
white crystalline layer observed in the present experiments. 
The black deposit consists of sub-oxides of sodium and 
potassium, and its appearance has been noted previously 
by the author *, although in rthat paper it was attributed to 
the hydrides. It has now been proved by chemical analysis 
that this black deposit does consist of the sub-oxides. 

Sulphur was tested in the following manner: — A small 
piece of filter-paper, soaked in lead-acetate solution, was 
placed together with a small amount of the solution in D. 
The rest of the apparatus was separated from D by a mercury 
cut-off not shown in the figure. C contained sulphur which 
had been deposited in a thin film over the interior. After 
exhausting the whole of the apparatus to a pressure of about 
5 mm. of Hg., the mercury cut-off between C and D was 
closed and the rest of the apparatus highly exhausted. 
Hydrogen was then admitted to A until the pressure was 
about 7 mm. Hg. While the electric discharge was passing, 
the mercury cut-off was opened. This was repeated many 
times, the pressure of the gas in A being gradually increased. 
Each time communication with D was established, any 
gaseous product formed in C was admitted to D. In the 
course of a few minutes the paper soaked with the lead- 
acetate solution turned black, showing the presence of a 
sulphide of hydrogen. This chemical compound must have 
been produced by the action of an active form of hydrogen 
on the sulphur. The surface of the mercury at the cut-off 

* Proc. Roy. Soc. A. vol. xc. (1914). 



224: Dr. F. H. Newman on Absorption of Hydrogen 

also lost its bright appearance. This was due to the action on 
it o£ the sulphide of hydrogen. The mercury surface 
remained quite clear when the sodium-potassium alloy was 
tested. 

Sulphur and the alloy were selected for tests because the 
chemical actions in these cases give rise to compounds whose 
effects can be noted easily. It is extremely difficult to 
examine phosphorus and iodine in this way owing to their 
high vapour-pressures. A possible test would be the com- 
parison of the vapour-pressures before and after absorption 
of hydrogen had taken place. 

These two experiments indicate that the chemical action is 
not due directly to the ions in the discharge-tube, as they 
were all eliminated by the charged platinum strips before 
reaching either the alloy or the sulphur. 

Wendt * showed that hydrogen can be activated by the 
passage of a-rays through the gas, and it has been shown by 
the author f that the active modification so produced is able 
to react chemically with sulphur and the alloy of sodium and 
potassium. 

4. Discussion of Results. 

The disappearance of gas in a vacuum-tube is due probably 
to several principles, some of which may be fundamental. 
It is certain, however, that any attempt to explain the 
principles by the same theory would lead to conflicting- 
results, but the processes occurring can be divided into 
two classes, chemical and mechanical. There is much 
evidence that the gas can be caused by the electric 
discharge to adhere to the solid parts of the discharge-tube 
in some manner which is at present unknown. In many 
cases a portion of the gas can be reliberated by heating 
the vessel, but no reason can be advanced for the non- 
reliberation of the whole of the gas which has disappeared. 

Langmuir assumes in the paper previously quoted that the 
hydrogen in the presence of an incandescent filament under- 
goes dissociation. The gas shows abnormal thermal 
conductivity at high temperatures, due to its atomic nature. 
The dissociation does not occur apparently in the space round 
the wire, and is not due to the impacts of the gas molecules 
against its surface, but takes place only among the hydrogen 
molecules which have been absorbed by the metal of the wire. 
Some of the atoms leaving the wire do not meet other atoms, 

* Nat. Acad. Sci. Proc. vol. v. (1919). 
t Phil. Mag. vol. xliii. (1922). 



by Elements in the Electric Discharge- Tube, 225 

owing to the low pressure, but diffuse into the tube cooled by 
liquid air, or become absorbed by the glass, and thus remain 
in the atomic condition. They retain all the chemical activity 
of the atoms. Langmuir also found that when the liquid air 
was removed, some of the atoms would come off the glass 
and recombine with other atoms to form molecules. These 
molecules could not be recondensed by replacing the liquid 
air. This gas which would not again disappear he termed a 
" non-recondensible" gas. 

This hypothesis, which is applicable to very low pressures, 
cannot hold at the pressures used in the present work. The 
gas in the atomic condition can scarcely move from the 
discharge-tube for a considerable distance and still retain 
its atomic nature. The " non-recondensible " gas found by 
Langmuir is probably hydrogen in its normal state. 

When nitrogen gas disappears in the discharge-tube, 
practically none of it can be reliberated, even when the tube 
is heated to the softening point. This fact indicates a 
striking difference between the disappearance of hydrogen 
and nitrogen. If chemical compounds are formed by the 
absorption of the gases, this difference can be explained 
in terms of the difference in the stability of the hydrides and 
nitrides produced. 

The chemical action may take place between hydrogen and 
the vapour of the element, and also it may occur at the 
surface of the solid. The majority of the elements studied 
have such small vapour-pressures that a very small propor- 
tion of the action is due to the vapour. The active condition 
of the gas must be caused by the ions, although results seem 
to indicate that the number of active atoms or molecules in 
the gas is of a much higher order than the number of ions 
present in the gas at the instant of recombination. 

The absorption is not due entirely to chemical action, as the 
law of constant proportions does not seem to be followed. 
It is of significance, however, that the rate of disappear- 
ance of the gas increases, and the final pressure attained 
decreases, as the temperature of the discharge vessel is 
lowered. This arises from the lowering of the vapour- 
pressure of the compounds produced, with the result that 
the final pressure reached is lowered. 

Although the formation of hydrogen sulphide in the 
discharge-tube by the action of the activated hydrogen 
on the sulphur will not explain the disappearance of the 
gas, it does indicate the production of a modified form of 
the gas which is able, possibly, to form other compounds 
with sulphur in addition to hydrogen sulphide. 

Phil Mag. S. 6. Vol 44. No. 259. July 1922. Q 



226 Mr. Bernard Cavanagh on 

That the mechanical deposition o£ gas on the walls of the 
discharge vessel will not account entirely for the disappear- 
ance of the gas is shown by the difference in the behaviour 
of nitrogen and hydrogen with phosphorus, sulphur, and 
iodine. Practically none of the nitrogen can be reliberated 
by heating, but a large proportion of the hydrogen is evolved. 

There is reason for believing that the modification of 
hydrogen is triatomic in nature. Wendt has shown in the 
paper previously quoted that hydrogen drawn from a tube 
through which an electric discharge is passing contains a 
small quantity of H 3 . Probably monatomic hydrogen is first 
formed, and owing to collision with neutral molecules of the 
gas, H 3 then appears. The monatomic gas may be produced 
originally by the action of the swift-moving electrons on the 
molecules. Wendt and Grubb * have also shown that N 3 is 
produced when an electric discharge passes through nitrogen. 
Thomson f found evidence of H 3 in his positive-ray experi- 
ments. It is this triatomic form of hydrogen which is 
effective in the production of chemical compounds in the 
electric discharge-tube. 



XVI. Molecular Thermodynamics. II. By Bernard 
A. M. Cavanagh, B.A., Balliol College, Oxford %. 

I. Molecules, Thermodynamics, and 
Quantum Theory. 

IN developing a molecular treatment of the thermodynamics 
of dilute solutions in simple solvents, Planck § deter- 
mined the form of the integration constants in the entropy 
function by a method which was at the time the subject of 
some controversy. 

M. Cantor || objected that the hypothetical transition to 
the gaseous state without change of the molecular composition 
was not even theoretically possible, since there probably 
existed in the liquid state, complex molecules whose existence 
was inseparably connected with the condensed state of the 
phase, and entirely incompatible with a state of high 
temperature and low pressure. 

* Science, vol. lii. (1920). 
t Proc. Roy. Soc. A. vol. Ixxxix. (1913). 
t Communicated by Dr. J. W. Nicholson, F.R.S. 
§ « Thermodynamics,' 1917 (Trans. Ogg), pp. 225-226. Or see Phil. 
Mag. xliii. p. 608 (1922). 

'I Ann. der Phys. x. p. 205 (1903). 



Molecular Thermodynamics. 227 

In reply, Planck * pointed out that the theoretical pos- 
sibility of the ideal transition depended only on the fact thai 
the numbers of the various molecular species were, together 
with temperature and pressure, the independent variables 
which determined the phase. 

Now the present author would suggest that Planck's reply 
can be construed (and, to be unanswerable, must be construed) 
as a rider to the definition of the terms " molecule " and 
" chemical compound," for the purposes of molecular 
thermodynamics, viz. : — " That the numbers of the various 
molecular species can be considered, together ivith temperature 
and pressure, as the independent variables determining the 
phase/'' or, in other words, " that it shall be theoretically 
sound to conceive any desired change of temperature and 
pressure of the system as taking place without change in the 
numbers present of the various molecular species f.." 

Oniy with this rider to our definition can it be laid down, 
for instance, that the "mass-action" equilibrium law must 
be obeyed in sufficiently dilute solution, for it is to be 
observed that purely " general " thermodynamics has no 
cognisance of molecules, but takes for its independent 
variables, besides temperature and pressure, the masses of 
the "components.'" [See next section of this paper.] 

The misconceptions which have so long stood in the way 
of a satisfactory general theory of electrolytic or "con- 
ducting " solutions seem sufficiently to illustrate the indis- 
pensability of this postulate. 

A parallel illustrating its signih'cance may be drawn from 
the dynamical theory of chemical combination and dis- 
sociation. 

The classical dynamical conception of a binary molecule 
(for example) was a pnir of simpler molecules (or atoms) 
moving relatively to one another in closed orbits, and the 
principle of the conservation of energy forbade the spon- 
taneous dissociation of such a " molecule," requiring that its 
disruption should depend on collision with another molecule. 

The well-known fact that dissociation is (at constant 
temperature) independent of collision-frequency, showed the 
inadequacy of this conception, and pointed to some property, 
in the "forces" producing and maintaining a molecule, 
altogether incompatible with the older or "continuous" 
dynamics. 

* Ann. der Phys. x. p. 436 (1903).^ 

f Intermediate and tinal states being unstable, of course, in general. 

Q2 



228 Mr. Bernard Oavanagh on 

The same difficulty arose when a dynamical explanation of 
the law of mass-action was attempted, the essential continuity 
of action of u physical " forces * standing in the way, and 
Boltzmann had to assume — with conscious artifici;tlity — dis- 
continuity in a field of force in order to arrive at the desired 
result. 

It seems indeed that, besides the more obvious and less 
peculiar properties of shortness of range, " specificity," and 
saturability, there is a quality of discontinuity of action (in 
time or in space or in both) which distinguishes "chemical 
forces '• from " physical," the distinction being sharp so far 
as we can yet see. 

We find then an absence of direct dependence of 
dissociation upon the thermal motion closely connected with 
the possibility of accounting for the law of mass-action 
dynamically. 

The parallel with our " rider " and its indispensability as a 
basis for the deductions of molecular thermodynamics is 
significant. 

The transitory orbital system which was the older 
" physical " conception of a molecule, and which quite 
probably occurs in all dense gases and liquids f, is clearly 
quite directly dependent on the thermal motion, is, in fact, 
itself merely an u episode " in that motion, and cannot in 
any sense be regarded as fulfilling the requirement of out- 
rider. We cannot, therefore, predict from thermodynamics 
the " mass-action " equilibrium law for the " reactions " of 
such " molecules " under any circumstances, for we cannot 
treat them 'as molecules for the purposes of molecular 
thermodynamics. And, in parallel, we find that dynamical 
theory is unable to predict the law of mass-action for such 
" molecules." 

The electrolyte question provides an important application 
of these considerations. 

It has frequently been supposed that a pair of ions, closely 
linked by their electrostatic fields alone, must be regarded as 
a molecule, and should behave thermodynamically as such. 
Electrostatic forces as we know them, however, are typical 

* That is, " forces " within the conception of the older physics. It 
seems convenient to use, in contradistinction, the term u chemical forces " 
for the " forces " or means by which a molecule is formed or /' hound " 
[see end of this section] and held together, and the expression may 
find some further justification in the fact that in the nature of these 
latter '' forces " lies, probably, the key to all the facts and phenomena of 
chemistry. 

t Compare here Cantor's objection, mentioned above. 



Molecular Thermodynamics, 229 

" physical " forces, and it is our present conclusion that such 
forces are capable of forming only transitory associations — 
' f episodes in the thermal motion " — essentially different from 
what we regard as molecules. 

\\ hen, as in the weaker acids and bases almost certainly, 
we really have partial ionization, or rather partial association 
of the ions to form u undissociated molecules," the latter 
must be regarded as produced and held together not by such 
ordinary electrostatic forces, but by " chemical forces " with 
the peculiar property already discussed. 

With regard to " strong electrolytes/' the work of Debye 
and Bragg oives good reason to believe that the molecule in 
the salt crystal is the ion. If this be so, it appears necessary 
to admit that the solid salt is essentially a mixed crystal whose 
special simplicity and homogeneity is due simply to the 
polarity of the electrostatic forces which dominate its 
" growth." 

Now, it would seem altogether inconsistent to suppose that 
the chemical " association " which does not take place in the 
intimate contact of the solid state, ensues when the ions are 
dispersed in a solvent, so that until the calculated effect of 
the electrostatic forces between the ions upon their thermo- 
dynamic behaviour can be shown to be inadequate when 
compared with experiment, the " complete-ionization " theory 
seems the only rational theory for strong electrolytes. 

Another application of the above general conclusions is to 
be found in the important question of " solvation " of solutes, 
which is treated in a paper to follow this. 

There occur in the literature of this subject such state- 
ments or suggestions as that " the solvates need not be 
definite chemical compounds," and vague theories of the 
" solvate molecule " as a mere indefinite conglomerate. 
From the preceding, at any rate, it is our conclusion that, 
unless the u solvate molecule " is produced and maintained 
by ' f chemical forces " in the sense already considered, so that 
it fulfils the requirement of our " rider," it will not, for 
the purposes of molecular thermodynamics, be a molecule 
at all. 

With regard to this remarkable characteristic of u chemical 
forces " which appears to be reduced to its lowest terms in 
the expression " discontinuity of action," this seems to mark 
out the problem of molecule formation (including, be it noted, 
reaction-velocity) as one of those many whose solution may 
be hoped for from the new quantum- dynamics of phenomena 
on the atomic scale. Indeed, it is tempting to believe that 
Bohr's conception of u electron-binding " may be the solution 



230 Mr. Bernard Cavanagh on 

in embryo o£ the larger and more complex problem o£ " atom- 
binding/'' and that in his distinction between "bound " and 
" unbound " electrons in the atom, we may have in its 
simplest aspect the distinction between "chemical" and 
" physical " forces. 

II. Molecular Thermodynamics. 

In general thermodynamics, which is based on, and applies 
to experience, the independent variables, besides temperature 
and pressure, are the masses of the "components/' and 
these are reduced to the minimum necessary to define the 
system under all circumstances not conventionally or prac- 
tically excluded from consideration. 

To take a familiar example, hydrogen and oxygen will 
suffice as the components of a system containing in addition 
water, provided low temperatures are excluded from 
consideration, or the presence of efficient catalysts is assumed. 
In so far as we may suppose that the decomposition and 
formation of water do proceed even at low temperatures in 
the absence of catalysts, though at an immeasurably small 
rate, it is clear that theoretically the two components would 
always suffice for this system if sufficient time ivere allowed. 

And conversely they would never suffice if the rate of experi- 
mentation were sufficiently increased. 

Striking examples of the practical reality of this entry of 
the time factor into the question of the necessary number of 
components, have been given in recent years by the work of 
A. Smith and of A. Smits, who by increasing the rate of 
experiment ation, have increased the number of components 
necessary to describe certain systems, the latter author having 
propounded an interesting theory of allotropy on the basis 
of his experiments. 

Now we can conceive this carried far beyond the bounds of 
purely practical limitations, and the question arises, " How 
far ? * 

The atomic or elementary theory of matter is introduced 
when we say that at one extreme, when unlimited time is 
available, the elementary atomic species will be necessary as 
well as sufficient as the components of any system. 

Starting from this extreme and increasing the rate of 
experimentation we can imagine one complex after another 
of these elementary atoms (as its rate of formation and 
decomposition ceases to be great in comparison with the rate 
of experimentation) taking its place in the list of " com- 
ponents necessary to describe the system." Remembering, 






Molecular Thermodynamics. 231 

however, that the masses of the components are together with 
temperature and pressure the independent variables, we see 
that there may be a limit, for it must be theoretically possible 
to alter temperature and pressure so quickly that the 
numbers present of these complexes which we are admitting 
as components remain sensibly unaltered during the change. 
According to the view put forward in the previous section, 
it involves us in a definite postulate bearing upon the nature 
of molecules and of chemical change, when we say that the 
limit is reached ichen and only ivhen every molecular species 
which can be formed in the system has taken its place in the 
list of ' ; components.''' 

Proceeding in theory to this limit, we obtain the general 
expression for -fy which is referred to in the sequel as the 
".molecular expression for ty" and as Planck showed, we can 
determine it completely when it is linear by " connecting-up " 
with the known properties of the low-pressure gaseous 
mixture. 

When the expression is not linear, the higher or " general " 
terms are subject only to a single limitation inherent in 
Planck's method, as pointed out in the previous paper *. 
Observing that the " general " terms in the corresponding 
expression for U f do not involve " chemical" energy, the 
present author also suggested and illustrated J the interesting- 
possibility of employing ordinary dynamical theory, at least 
as a valuable aid, in determining and interpreting the form 
of these " general " terms. 

This "molecular expression for yfr," however, will clearly 
not in general correspond with our experiments carried out 
under ordinary conditions. They will correspond with an 
expression of the "general thermodynamic" type in which 
the components are appropriate to the conditions of ex- 
periment. 

The theoretical problem then presents itself of connecting 
this " experimental" expression for -\jr, in a manner at once 
rigorous — that is trustworthy — and practically effective, with 
the "molecular" expression and its possibilities of theoretical 
interpretation. 

The treatment of two important problems of this kind has 
been attempted. 

Planck pointed out that when a single molecular species 

* Phil. Mag. xliii. p. 606 (1922). 

t Entirely analogous considerations apply to V, but ordinarily owing 
to the low pressures used, V figures relatively negligibly in the determi- 
nation of ^ [c/. footnote, p. 630, Phil. Mag. xliii. (1922)]. 

X Loc. cit. p. 62"). 



232 Mr Bernard Cavanagh on 

greatly preponderates, it is a matter of mathematical 
necessity that the " molecular expression for \jr " should take 
(in the limit) a linear form, and it was to this type o£ 
" dilute solution " that Planck confined himself, arriving 
readily at the Raoult-van't Hoff " laws of dilute solution." 
Dilute solutions in a liquid paraffin would be of this 
type. 

Van Laar * took the linear expression as the criterion of 
" perfect solution 5 ' in general, and not making the approxi- 
mations which Planck, considering very low concentrations, 
had made, was able to show that the Raoult-van't Hoff 
laws formed too restricted a criterion when the solution was 
very dilute. 

He, however, considered only solutions in which the 
solvent was of the same type as that of Planck, viz. : a 
single molecular species. 

In view of the fact that the Raoult-van't Hoff laws have 
been found to hold for dilute solution in our common and 
useful solvents, which are certainly not of the type considered 
by Planck and Van Laar, the present author was led to the 
problem of " complex solvents," which will be the first illus- 
tration of the theoretical problem outlined above. A 
preliminary treatment appeared in the first of these papers, 
but a more complete and rigorous treatment is now presented. 

The second illustration will be the problem of partially 
" solvated " solutes, a discussion of which will follow that of 
" complex solvents." 

The first result is that the Raoult-van't Hoff laws have 
been rigorously predicted for extreme dilutions in such 
solutions. It is shown, in fact, how the " experimental " 
expression for i/r simulates, in the limit, the " molecular " 
expression in form. 

But farther the way is prepared for the thorough investi- 
gation of middle and high concentrations in such solutions. 
To this end the "linear" terms in the experimental 
expression for i|r, which simulate and replace the simple 
linear terms in the molecular expression, have been treated 
with some thoroughness and rigour, these being the terms to 
which the expression reduces when the solution is " perfect." 

When the quite practical criteria thus provided are applied, 
the belief that " perfect solution " always ceases in these 
"complex" solutions at quite low concentrations may be 
largely dispelled. 

In simple solutions of the kind considered by Planck and 

* Z.f. Phys. Chem., several papers, 1903 etc. 






Molecular Thermodynamics. 233 

Van Laar we find "perfect solution" persisting up to very 
high concentrations — sometimes over the whole range. 

The "general" terms of which (excepting the case of 
electrolytes) little or nothing is yet known, have been touched 
on only in so far as the treatment of the " linear " terms 
involves (in general) a certain very slight alteration in the 
division into u linear " and "general" terms, which may 
sometimes have to be taken into account when dynamical 
theory is employed. 

Experimental determination of the " general " terms for 
comparison with theory must of course be preceded by 
knowledge of the " linear " terms, — hence again the need 
for rigorous and thorough treatment of the latter. 

III. Complex Solvents. 

The importance of this question of " complex" (polymer- 
ized and mixed *) solvents is sufficiently obvious when it is 
considered that of this type are most of our best solvents, 
probably all our " ionizing " solvents, and, chief of all, water. 

We have to consider a solution consisting of the solute- 
molecular species, m l3 m 2 , , m s , . . . . , in addition to the 

various species m 01 , m 02 , . . . . , which constitute the solvent. 

Concentrations being expressed in gram-molecules per gram 
of solvent, the " molecular " expression for yjr is 

+ S» s (&-Rlog^-J^-) +-RM S4, x 'f x (e 01 ... ,....), (1) 
which is of the form 

*-*"%+HWj- ■■ ■ > 2 > 

the several solvent-molecular species appearing as separate 

* Mixed solvents, while submitting to the same theoretical treatment 
as the merely polymerized solvents, present certain peculiar difficulties 
and some interesting possibilities with which it is hoped to deal at length 
in a later paper. 

t A suffix outside a bracket is used to indicate, in less obvious cases, 
independent variables which are held constant in a partial differentiation, 
— a well-known usage. A single suffix may be used briefly for a whole 
series, as n 0] here standing for w i, n 2, 



234 Mr. Bernard Cavanaoh 



on 



components, whereas the " experimental '"' expression must 
have the form 

t = M„|J + 2 ras (|t) ..... ( 3 ) 

the solvent appearing as one component only. 

Clearly 

(*±\ = (^±) + f S ^r_ .3^oi] , . (4) 
\'dn s l m \dn,/n 0l L drc 01 ' dn s jM ' 

but the relation we have to use in making the change of 
variables is that given by the condition for chemical equi- 
librium among the molecules of the solvent, viz. : — 

rs|* .dHoii =o, (5) 

L On 01 J Mo 

so that 



\dn s J Mo \bn s / noi 



(6) 



and comparing then (2) and (3), 



BM;" Scol d^' (7) 

which could be regarded as physically obvious, as was done 
in the preliminary treatment (previous paper). 

We shall abbreviate (1) by writing m for = — ; C for %c s , 
and M G' for the " general " terms> so that, 

t = 2»o 1 (*oi-Rlog I ^| D ) 

+2» s (<#, s -Rlog I ^| u )+M G'. . (8) 

It will be convenient also to write G / for ^ — (M GT), 

the "general ' terms in s~^ ; similarly GJ for those in 

|i, etc. 
dn s 

Now, in the first place, we have to show that it is permis- 
sible to assume that (8) has already been so arranged that 
Gr', G s \ Goi', etc., all vanish in the limit when C becomes 



Molecular Thermodynamics. 235 

very small. This is essential to the rigour of the demon- 
stration that the Raoult-van't Hoff laws are still the limiting 
laws o£ dilute solution when the solvent is complex. 

Now G / is, at constant temperature and pressure, a func- 
tion of c i, c 02 , . . . . , as well as of e l9 c 2 , Owing to the 

chemical equilibrium controlling coi, r 2, • • • • , liowever, these 
quantities have, in the pure solvent, values depending only 
on temperature and pressure, and the departures from these 
limiting values, caused by the presence of solutes, will clearly 
decrease with the concentrations of those solutes. Thus as C 
diminishes, the ranges of variation to be considered of the 
variables cbi, C02, «... are progressively limited, the same 
being obvious in the case of c u c 2 , 

But clearly any finite, continuous, ditf'erentiable function 
of several variables must behave as a linear function if the 
range of variation considered of every variable is sufficiently 
limited *. 

Thus 

LtG' = 2c iZoi + 2cA 5 ( 9 ^ 

C-yO 

i. e. 

Lt(M G')=Woi+2W,, • • • • C 10 ^ 

where Z 01 , , ? 1? , depend only on temperature and 

pressure, being, in fact, limiting values of Gr 01 ', , 

G/ , respectively. 

But clearly 7 i» > ^> , can be transferred to, and 

included in the " linear " terms, — in O1 , , fa, re- 
spectively — whereupon the residual " general " terms will 
satisfy the requirement that G', G s ', G 01 ', etc., should all 
vanish as C becomes very small. We shall assume that 
in (8) this adjustment has already been carried out. 

Returning now to (7) and comparing with (8), we see that 

C>OO iU o 00 L OU 0l J 

= It ["icoiC^oi-R^g"^!] I = 0M ( saY ); ' ( U ) 

since, in the pure solvent, c ol , and 777 assume values 

dependent only on temperature and pressure. 

* Merely the obvious property of tangency in re-dimensions. The 
theorem quoted by Planck in treating simple solvents ['Thermo- 
dynamics' (Trans. Ogg), 1917, p. 225] appears to be the particular case 
of this, when the ranges of variation of the variables are all located (as 
here in the case of c 1} c 2 , . . . .) close to zero. 



236 Mr. Bernard Cavanaffh 



i=> 



on 



Then we have 



but again remembering 

Sj£dta=0, (13) 

O n oi 

and so from (8) 



'^ffr*jXpn(~Bdlogg$fe) + fw«V 



(15) 



c=o c=o 



= <£ M + R(idlog(l+mC-t- jScoirfGoi. ■ (16) 



(17) 



c=o c=o 

Now (6) with (8) gives 

\B^/Mo L \<Wn J, ft H-mO 

And if we write (16) in the form 

^^M+nfirfi.og (i+mC)+a M , . (is) 

*c=o 
it is at once clear that the " general " terms thus adopted 
(and therefore, of course, the " linear " terms similarly) are 
connected by the Gibbs fundamental relation [see note at 

end of this paper], for (M G') being a function of n 0l 

^i , homogeneous and of the first degree, 

Z>ioidG ol ' +2n s dG s ' = ; 

i-e. -M Zc 01 dG Q1 ' + Xn s dG s ' = 0, 

i-e. Mo^Gm 4-2n,dG,' = 0*, . . , (19) 

which means that [M G M + %nJ3t s '} or (say) M G is, as a 

* The Gibbs relation might indeed have been used to obtain (16) direct 
from (17), <j) M appearing as the integration constant. The above treat- 
ment [(12) to (16)] appeared, however, to be more interesting, and to 
introduce M in a more natural and illuminating manner, in its relation 
to the original " molecular " expression for ^r. 



Molecular Thermodynamic 



237 

function of M and n^n^ , homogeneous and of the first 

degree. And the same will hold for the " linear " terms, (if 
(M G) be accepted as the new " general terms"), a point of 
obvious importance since the "linear"" terms must alone 

remain when the solution is " perfect." ^ retains the 

on s 

simple form (17), which may now be written 



(20) 



since 



d(M G)\ BG 






(21) 



G being a function of c x c 2 only (besides T and p), 

Also, of course, 

122) 



\ B» S /noi ' 



and 



M G=M,G M + S«,G S 



(23) 



We observe, however, that this convenient arrangement 
involves theoretically a certain definite (though probably 
always very small) change in the division into " linear " and 
" general " terms, since 

M G' = 2h 01 G i' + n sG s / ; 

G'=Zc 0l G 0l ' + Xc s G s 

= G+f 2G 01 '^ 01 (21) 

~c=o 
Xow it is readily shown that 



Grni' = 



d^oi 



i [n< v„ ^G v ^G n 
+ m i G — ZCoi^-, 2.<?* ^— 



(25) 



and since, 2m 01 c 01 being unity, 2??z 01 <ic 01 is always zero, we 
have 



G 



'— g= (v ?^*- 

J 3<?oi 

c=o 



(20) 



To say that any such modification of the division into " linear " 
and "general" terms is due to, and represents, a departure 
on the part of the solvent molecules from " perfect " 
behaviour would be to make a qualitative statement of no 



238 Mr. Bernard Cavanagh on 

practical significance. (24) is a quantitative statement, and 
the form to which it is reduced in (26) has practical meaning 
and value, as will be shown by means of a simple illustration 
in a short appendix to this section. From (26) it is clear that, 
owing to the relative smallness of the changes in c 01 , c 02 , 

produced by the presence of solutes, the difference 

between the "general" terms in the original " molecular " 
expression for -v/r and those in the " experimental " expression 
we have obtained will generally be very small. But it may 
have to be taken into account in making use of dynamical 
theory (at high concentrations). 

Of course, until and except when the "general" terms 
can be given more definite form, we cannot say anything 
about the way in which they will depend on the constitution 
of the solvent and its variation . For the present we have to 

suppose that c i c 02 are eliminated in terms of T, p, c 1} c 2 , 

s from Gr, which takes some form 

G = UXcp x f f x (c lC2 ) (27) 

Theoretically, and in the general case, the application of 

dynamical theory will precede this elimination of c 0] c Q2 , 

will deal in fact with the original general terms M G-', so 
that, in greater or less degree, knowledge of the constitution 
of the solvent and its variation will be necessary before such 
theory can join issue with practice, but in some cases, as the 
appendix will illustrate, this may not be necessary, even 
though the constitution of the solvent does affect the 
" general " terms. 

It should be noticed that the value and convenience of our 
" experimental " expression for -\|r is by no means entirely 
dependent on a deficiency of knowledge of the constitution 
of the solvent, though the latter makes it practically 
indispensable. 

In the " linear " terms the effect of the constitution of the 
solvent is concentrated in the quantity m, the mean molecular 
weight of the solvent. 

In the pure solvent this will have a limiting value m 
dependent only on temperature and pressure, and we can 
therefore regard the quantity ( — R log m Q ) as included in the 
quantity <f> s , when our " experimental" expression for yjr will 
finally take the form 

^ = M [<£ M + R f = dlog (1+SC)] 

*c=o 
4-Sn s [^-R{logc 5 -log(| +m c|]+M G . . (28) 



Molecular Thermodynamics. 239 

when m is constant (at constant temperature and pressure) 
and therefore equal to m we get 

f=M„|> M + Jlog (1 +-, C)1 

+ 2wJ^-RJ logc,-iog(l + m O)T]+M G, . . (20) 

which is equation (52) of the first of these papers, from 
which the " second approximation " equations were obtained. 

It is, of course, not possible to say how, in the most 
general case, m will depend on the concentrations of the 
various solutes, but an interesting case, of probably very wide 
application, may be treated and will at the same time serve 
as an illustration. 

This is the case where m can be written, with sufficient 
approximation, as a series of ascending integral powders of C, 
the total solute concentration. 

This can be shown to be the case, for instance, when the 
various solvent-molecular species behave as perfect solutes 
(in the true sense, — not in the sense of the Raoult-van't 
Hoff laws). 

Some simple cases have been investigated, but the detail 
need not be given here. It will suffice to say that in the 
simplest case, for example where the solvent consists of two 
molecular species, m and (2 m ), the one the doublet of the 
other, we find that ~m can be expressed as 

m = m [l + 6{m C) + V (m C) 2 ], 

where the values of 6 and y depend, of course, on the 
proportions in which the two species are present in the pure 
solvent, but in any case cannot exceed §• and -3-g respectively 
(these maxima not being simultaneous). 

We may carry this expansion of m, which is formally 
convenient for our purpose, to one further term of which 
only the order of magnitude will matter, 



in 



.=l + 0(- Q C) + V (m Cy + Z(m C^ . . (30) 



the last term being, as we shall find, altogether negligible if £ 
is no greater than about y 1 ^. 

Approximating on the assumption that 6 and r) are of the 



240 Mr. Bernard Cavanagh on 

order of magnitude ^ we obtain : 

log^+woO^aiO-iajC + JaaC 8 , . . (31) 
| J^log(l+mC)=C(l-ia 1 C + ia 2 C 2 -ia 3 C 3 ), (32) 



c=o 



where a 1 = ?w (l — 0), 

a 2 = m 2 [l-2(<9-7 7 )], ... (33) 



a 3 = m o 3 [l-3(0-77 + f 



it being clear that, as stated above, £ can be neglected 
altogether if no greater than about fa 
We get then the equations 

( W =^ M +RC[i-KC+^ 2 c 2 -ia 3 ^] + g m j 
IV - U*° (34) 

|i- =^-R[log<; s -a 1 C + ia 2 C 2 -ia i) C 3 ]+G s 

and thence the successive stages : — 

U=<£ M +RC[Wmo(l-0)C + ; WC 2 ] +G M ) 
U/° (35) 



^=* M + RC[l-^ O] + G M 



and 

at 

,3M, 
II. . .- .... (36) 

||£ = *.-R[logc.-m C]+G, 

and 

(|±=^ + RC + G M j 
I. * M ° ....... (37) 

f|£ =^-Rlogc, + G, 

\ 0^9 J 

I. and II. being the first and second approximations 

obtained by the preliminary treatment in the previous paper. 

Taking I per cent, as the " probable experimental error," 



Molecular Thermodynamics. 241 

we get roughly the following upper limits (of total concen- 
tration C) for the applicability of the four successive stages 
of approximation : 



Approximation I 



M. 

10 

II 2M. 

Ill 5M. 

IV 8M. 



(38) 



— that is, considering aqueous solution, and assuming m to be 
about 40. The limits would be considerably different, par- 
ticularly in the case of III. and IV., if m were given a very 
different value, as can readily be seen. 

The practical criterion of a perfect solute in a complex 
solvent is now that its behaviour should be expressed by that 
one of the above succesive approximations appropriate to the 
total-solute concentration of the solution, with the "general " 
terms omitted. 

If the assumption of perfect behaviour in the case of a 
particular solute be made, an experimental determination of 
the quantities ra , 0, rj, etc., can be made and concordance in 
several such determinations made upon different solutes 
would tend to justify the assumption that perfect behaviour 
persisted up to the concentration at which concordance was 
found. 

According to the concentration reached (with concordance) 
some of the quantities m , 0, 77, etc. would then be known 
with some approximation (closest in m , next in #, and so on). 

Then, on the assumption of "perfect" behaviour on the 
part of the solvent-molecular species — that is, a sufficient 
approximation thereto, — these quantities would suffice to 
discover something about the constitution of the solvent. 

Thus two solvent-molecular species would be completely 
determined by a knowledge of m alone (that is, the propor- 
tions of the two kinds present would be determined), — and 
in this case the question of "perfect" behaviour of the two 
species would not enter. m and 6 would, suffice to deter- 
mine completely three species, or would provide a check if 
only two species were present, and so on. 

The problem, however, as a practical problem is compli- 
cated by considerations which are the subject of a succeeding 
paper, viz. the question of solvation and partial solvation of 
solutes. 

Finally, it is proposed to consider one point with regard 
Phil. Mag, S. 6. Vol. 44. No. 259. July 1922. R 



242 Mr. Bernard Cavanagh on 

to the expansions of U and Y, the total energy and volume 
o£ the solution. (Fuller consideration is postponed, as this 
paper is already rather long.) 
In " perfect " solution we have 



U = %n 0l u 01 -f %n s u 
and 



■„ } • • • ■ (39) 

+ 2<n s v s J 



V = %n 0l v 0l 

Q = %n 01 q 01 + %n s q s , (40) 

where, according to the usage of the previous paper, Q is 
(U+joV), q s is (iis+pvs), etc., and we can write this 

Q = M 2coi?oi+2w,ft (41) 

2coi#oi> however, depends (through c 01 , c 02 ; . . . .) upon the 
concentrations of solutes present, but has, in the pure 
solvent, a limiting value <^m depending only on T and p, and 

tc 01 q 01 = q M +%qoi\ dcoi . • • • (42) 

Jc=o 

= q M + %q 01 Ac 01 (43) 

And so 

Q = M ? M +2w^ + M SgoiAc i. . • ■ ( u ) 

Dilute now such a solution "infinitely" by adding a large 
mass M(/ of pure solvent at the same T and jo, for which 

Q' = M '^M. 

The united heat will be 

(Q+Q') = (Mo + MoO^M + Sw^ + MoSgoiAcoi. 

But since the solution is now " infinitely " dilute, we shall 
have on bringing it to the original temperature and pressure 

Q"= (Mo + M ')?m+2m^= (Q 4 Q / )-M 2?oiAc i. 

In other words, MqX^oi^^oi was the heat developed (evolved) 
on diluting the %n 8 molecules of solute. 
That is, there is a heat of dilution of 

ytq 01 Ac 0l (45) 

per gram-molecule of solute, in " perfect " solution in a 
" complex " solvent. The explanation of this apparent 



Molecular Thermodynamics. 243 

anomaly is that in a "complex'' solvent the process of 
dilution is not quite so simple as in a "simple" solvent, 
being accompanied by a change in the constitution of the 
solvent — a reversion, in fact, to the constitution of the pure 
solvent. It is, of course, plain that M 2<7 01 Ac 01 , or SgfoAwoij 
is the ("chemical") heat evolution accompanying this 
reversion (at constant temperature and pressure). 
With (44) analogously we have 

U = M m m -» Xn s u s + M ti( Ac 0U . . . (46) 

V = M vm + Ws + M XiWW, • • • (47) 

and it may easily be verified from (LI) that 






(48) 



Appendix to Section III. 

A simple example, which is essentially merely illustrative, 
but may possibly be something more, will serve to make clear 
the practical significance of equation (26) concerning the 
slight modification of the " general JJ terms. 

In section V. of the first of these papers was obtained for 
the general term in the case of a dilute solution of a binary 
strong electrolyte 

M G'=RM <£V /2 , (49) 

where (// depended, in some way, upon a certain " effective" 
dielectric constant (D), which, at sufficient dilution, would 
be that of the pure solvent. 

Let us suppose that D depends on c s in such a way that <f>' 
is a linear function of c s 

4>'=4>o'(l + ac,) 9 (50) 

and also, in the first instance, that this effect of c s on D is 
entirely independent of its effect on the constitution of the 
solvent, that is, that the slight change in the latter produced 
by the addition of the electrolyte would alone [if produced, 
for instance, by some different solute, c„ the concentration 

R 2 



241 



Mr. Bernard Oavanagh on 



of the electrolyte being made quite small] have but negli- 
gible effect on D, then G' is independent of the constitution 
of the solvent (practically), i. e. 



so that, 





i 


(Xoi 

=0 


= 0; . . 




G = 


G / 


= UCJ> 'C^\1 + UC S ) 


1 
* 


G s = 


G-J 


= lR<£o' c * 


1/2(1 +.5 aCg ) 


>)■'■' 


Gj\f= 


Zc 01 G i 


'=— 2^0 


'cv 3 / 2 (l-f3ac s 



(51) 



(52) 



If, on the other hand, we suppose that the effect of c s on D 
is entirely dependent on the change produced in the consti- 
tution of the solvent, and would be fully obtained if the 
latter could be brought about in some other wa}', while c s 
was made very small, then we should have 



an( 



2 dGT ^ Bcoi _ Rc 3/2 3<£! 

= Rc s 3 / 2 <£'a J 



c=o 



so that 



G'-a = |R(/) w s 



5/2 



(53) 

(54) 

(55) 



i. e. in this second case, 

G = G' -J 2|£'^ 01 =E</, V /2 (1+I«,) 

g s = g; = i r^ v /2 (i +«<■») 



(56) 



Gm = 2<"oiG 



-1 



BG' 



d^oi 

c=o 
In both cases, of course, 



-<Z< 



-iRtf> V /2 (l + ^) 



G s = G,/, 



but they are not the same in the two cases because in the 
second case <f>' is a constant when w i> n 02 . . . . are held con- 
stant, but not in the first case. (For the same reason 
ScoiGoi' is a l- different in the two cases.) 

In the second case we note that although the constitution 



Molecular Thermodynamics. 245 

of the solvent enters into the "general" terms it does so 
only through the quantity D, and if this is the dielectric' 
constant of the solution in bulk it can be. measured and so 
determined as a function of c s without considering the corre- 
sponding variation in the constitution of the solvent, or the 
way in which the latter exerts its effect on D. 

If, as is probable at the less extreme dilutions, D is not 
the experimental or " bulk " dielectric constant, but a certain 
statistical-average quantity of a peculiar kind, then its varia- 
tion is, at least partially, not due to a variation in the solvent, 
but directly to variation of c s as in the first case above. 

Clearly from the preceding it would theoretically be pos- 
sible iu such a case to determine from comparison of theory 
with experiment whether the effect of c s on D was direct or 
indirect or in what proportion both, but it might not be 
practicable owing to the smallness of the effects to be 
measured. 

Note on the " Gibbs Fundamental Relation." 

Consider any property it of a homogeneous substance or 
phase, which is determined in magnitude by the composition 
of the phase and the quantity of substance considered. 
In view of the homogeneity it must then be proportional to 
the quantity of substance when the composition is fixed. 

Such properties are (at constant temperature and pressure) 
U and V, the total-energy and volume, Q or (U + pY), 
which might be called the "reversible heat content," and 
any thermodynamic potential such as entropy, free energy, 
Gibbs' " chemical potential," or Planck's yjr, which may all 
be expressed (at constant temperature and pressure) as func- 
tions of the quantities ]M\ M 2 .... of the constituents which 
suffice, under the conditions considered, to produce the 
phase. 

We can show, as Planck does in the case of i/r, that for 
any such property it, 

- = SM W • • • • • («) 

for if e be some infinitesimal fraction and we remove (eMj) 

of the first constituent, clearly it is diminished by eMx ^rr- . 

Removing simultaneously the same fraction of the total 
quantity of each constituent we diminish w, in all, by 

SeMx^r^-, or eSMj— ^. But, in so doing, we have 



246 On Molecular Thermodynamics. 

simply removed the fraction e of the whole phase, without 
altering its composition, so that tt must have diminished by 
en, that is 

whence (57) follows. 

When tt is either yjr, or Gibbs' "chemical potential," a 
whole system of phases in equilibrium can be considered 

together, since then ^^ , etc., are the same in every phase. 

Without actually quoting Euler's theorem, Planck remarks, 
in regard to yjr, that this relation means that yfr, as a function 
of M : M 2 . . . ., is homogeneous and of the first degree, though 
not, of course, in general linear, and the same remark applies 
to our typical property tt. 

From (57) we can at once get a more practically useful 
relation by differentiating both sides fully:-— 

that is, 

MS)* (58) 

Now equation (97) of Gibbs' classical paper reduces at 
constant temperature and pressure to 

Xm^dfjbi = 

and is then simply (58) applied to Gibbs' " chemical 
potential." 

(58) is therefore referred to in these papers as the 
" Gibbs fundamental relation," but its general applicability 
to any property of the type of it (for a single homogeneous 
phase) is to be borne in mind. 

It is to be observed that while the constituents whose 
masses are M 1 M 2 .... must be sufficient to produce the 
phase under the conditions considered, they need not be all 
necessary — they need not be the " general-thermodynamic " 
components. And since also (obviously from the form of 
(58)) MiM 2 . . . . need not be expressed in the same units we 
see that equally valid is the form 



w (£) 



= 0, (59) 



the " molecular-thermodynamic " form, in which n x n 2 . 
are the numbers present of the various molecular species. 



The Calculation of Centroids. 



247 



It is an important point in the treatment of the two 
problems, "complex solvents " and "solvation," presented 
in this and a succeeding paper, that in the " practical " or 
"experimental" expression for yfr finally obtained the 
"linear terms" by themselves satisfy the Gibbs fundamental 
relation, for in perfect solution these terms alone remain. 
And this is preserved in the successive approximations. 
The relation also serves as a useful check upon the correct- 
ness of the detail. 

Balliol College, 
March 1922. 



XVII. The Calculation of Centroids. By J. G. Gray, D.Sc, 
Car gill Professor of Applied Physics in the University of 
Glasgow *. 

THE position of the centroid of a plane arc or area is 
usually determined by the application of one or other 
of the two theorems of Pappus. The methods described and 
illustrated below seem, however, to be novel ; they are useful 
in a great number of cases, including many to which the 
theorems of Pappus do not apply. 



Fi<r. 1. 



Fig. 2. 





Consider a system made up of two masses M and m 
(fig. 1) . Let the centroids of m and of the system M and m 
be at a and G respectively. Now suppose the mass m moved 
so that its centroid is brought to a'. G moves to G', where 
GG' is parallel to aa' ; and we have (M + m) GG' '= m aa' . 

As a first example, consider the case of a circular arc AB 
mass per unit of length m (say). Let be the centre of the 
circle of which the arc forms part. Now suppose the arc 

* Communicated by the Author. 



248 Prof. J. G, Gray on the 

rotated about in its own plane through a small angle 0, 
so that A is brought to A' and B to B'. In effect the small 
portion AA' of the arc is transferred from one end of the arc 
to the other. The mass of this element is mrd, and it has 
been translated (virtually) through the distance 2r sin a/2, 
where a is the angle AOB. If G is the centroid of the arc, 
we have obviously 

mrO 2r sin a/2 = mra OG 0, 



or 



0G = 



2r sin u/2 



As a second example we take the case of a sector of a 
circle OADC (fig. 3). Let the sector be turned in its own 
plane through a small angle 6 about an axis through 0, so 
that A comes to A', B to B\ The effect of the rotation has 
been (virtually) to transfer the triangle OAA' to OOC. 



Fig-. 3. 





The centroid G of the sector has moved parallel to gg' 
through a distance OG 6. The mass of the sector is JrW, 
and that of the triangle OAA' is ^r 2 6cr, where a is the mass 

of the sector per unit area. 



Since aa' = k t sin « , we have 



99 



-r 2 6<r - r sin - = - r 2 u<r OG 0, 



or 



OG = 



4 r sin a/2 



Again, let it be required to find the position of the centroid 
of a segment of a circle ABC (fig. 4). The segment is 
turned in its own plane, about 0, through a small angle 0. 
A is thus brought to A f and to 0\ If the mass per unit 
of area of the segment is cr, the mass of the triangle DAA' 



Calculation of Centroids. 249 

is o-(9^?'sin|a rsinja, or ^o-tfr 2 sin 8 %ct. The area o£ the 
segment is |r 2 « — r 2 sin -|acos J«, and its mass is 

cr/' 2 (-|a — sin \ol cos Ja). 

Since <7</ is v|?'sin-^«, we have 

o-? ,2 (-|« — sin Ja cos \c*)x6 = §o-0?< 3 sin 3 £a, 

or n 



a? = 



?' sin 3 i« 



3 ^a — sin ^acos-^a* 



where a? is the distance of the centroid of the segment from 0. 

Consider next the solids obtained by dividing a right 

circular cylinder into two parts by means of the plane abed 

(fig. 5). Let it be required to find the position of the 



Fis:. 5. 




centroid of the lower solid. We suppose the solid rotated 
through a small angle 6 about the axis 00' (the axis of 
figure of the complete cylinder) ; a is thus brought to a', 
b to b', c to c', and d to d' . In effect the wedge ebb'e'cc' has 
been removed from the solid and replaced in the position 
eaa'edd'. If A A denotes an element of area in abed at a 
distance x from ee\ the volume swept out by this element 
due to the turning of the solid is AAxO. The mass of this 
element of volume is p/\Ax0, and since the element is moved 
(virtually) through a distance "2x, we have, if V is the volume 
of the solid, 

YpOUe = 2pdZ AAx\ 

where the summation is made over the complete area abed. 
Hence 

V x OG = AK 2 , 

where A is the area abed, and K is its radius of gyration 



250 



Prof. J. G. Gray 



on 



tin 



about ee'. If a is the angle bOa, and I the length of the 
solid, we have for the sectional area 

\t 2 {2it — ct^ + r 2 sin -^a cos \a. 

And since A = 2lr sin ^a, and K 2 = %r 2 sin 2 -Ja, we have 



OG 



r sin 3 -ia 



IT- 



^a + sin ^acos \a 



Similarly, if G' is the centroid of the upper solid, we have 

ia — Sin fa COS fa 

The positions of the centroids of the solids obtained by 
dividing a right circular cylinder into two portions by means 
of planes Oa, Ob (fig. 6) are easily determined. If G is the 
centroid of the larger solid, we obtain at once, by supposing 

Fig. 6. 





the solid turned through a small angle 6 about the axis of 
figure of the complete cylinder, so that a arrives at a', and 

b at &', . 

VxOG = 2AK 2 sini<*, 

where A is the area represented by Oa, and K is its radius 
of gyration about the axis of figure. We have, if I is the 
length of the cylinder. 



OG 



2ZrJr 2 sin£a 
P(2fl—a)7 

4 r sin ia 



3 2ir-cc ' 
If the cylinder is divided into two equal parts, we obtain 



Calculation of Centroids. 251 

the position of the centroid of each by putting- a = 7r. Thus 

0G=^. 

O 7T 

For the portion of a sphere shown by the firm lines of 
figure 7, we have, if OG is the distance of the centroid from 
the centre of the complete sphere, 

where A is the area abed, and K is its radius of gyration 
about its diameter. Thus 

OP — 71 " r2 S ^ llS 2 a 4 r2 Sni2 2 U 



3 r 



For the portion of a sphere enclosed by the firm lines of 



figure 8 we have 



,2 \r 2 sin | 
x = 



•nr 4 *?' sin ka. 



o 2lT — OL 
7TT 6 



3 2tt 

3 irr sin \a. 

3 27T — OC 



where x is the distance of the centroid from 00'. For a 
hemisphere a = 7r, and 



x = i r. 



Finally, for the portion of an anchor ring shown in figure 9 
we have, if r is the radius of the ring and a the radius of 
the section, 

OP — 7ra,2 (i a 2 + 7 ' 2 i ^ sin £* 
7ra J X 27rr X 

Z7T 

_ 2{r* + ja*) sin Ja 
y(27T~a) ' 



in 



which reduces to 2rjir when a = 7r and a is very small 
comparison with r. 

If a body is floating partly immersed in a liquid, the 



252 Messrs. Trivelli and Righter on Silbersteins 

distance of the centre of buoyancy B (the centroid of 
the displaced liquid) from the metacentre M is given by 
BM = AK 2 /V, where V is the volume of the displaced liquid 
and AK 2 is the moment of inertia of the water-line area 
about the intersection of the wedges of emersion and im- 
mersion. The equilibrium of the floating body is stable if 

Fig. 8. Fig. 9. 





M is above G, the centroid of the floating body, that is 
if BM > BG ; it is unstable if BM < BG ; and it is neutral if 
BM = BG, that is when M coincides with G. In figure 5 we 
may suppose the complete cylinder floating in water so that 
abed is the water-line area. The cylinder is obviously in 
neutral equilibrium so far as turning about the axis 00' is 
concerned. Thus M lies in the line 00', and we have 



0G=^ 



University of Glasgow, 
May 1,1922. 



XVIII. Preliminary Investigations on Silbersteins Quantum 
Theory of Photographic Exposure. By A. P. H. Teivelli 
and L. Righter*. 

Introductory. 

THIS paper is the first of a number of investigations 
now being conducted in this laboratory to test experi- 
mentally the light-quantum theory of photographic exposure 
recently proposed by Dr. Silberstein before the Toronto 
meeting of the American Physical Society. 

* Communication No. 141 from the Research Laboratory of the 
Eastman Kodak Company. 



Quantum Theory of Photographic Exposure. 253 

Originally, these experiments were started independently 
of that theory, onr intention being primarily to study the 
effect of the clumping, or clustering together in groups, of 
silver halide grains in a photographic emulsion *. According 
to Slade and Higson f, " It seems reasonable to assume 
that each grain acts quite independently and that one grain 
which has become developable is unable to make a grain, 
situated in close proximity to it, developable unless the 
latter grain is developable in itself." From the table 
(p. 256) it is readily seen that this statement is not true, but 
the grains when clumped together act as one grain for 
development to the limit. 

Since each clump acts as one grain a very much broader 
range of grain sizes, or their equivalent, is obtained extending 
from the smallest single grains up to the largest clumps 
(containing 30 or more grains) in a given emulsion. 

It was thought that these results afforded a rigorous test 
for Silberstehrs theory, and it seemed therefore worth while 
to compare them with the implications of that theory. 

Silberstein's fundamental formula is essentially, i. e. apart 
from chromatic complications, and disregarding the lateral 
dimensions of the light quanta, 

& -j -na 

where N is the original number, per unit of the p'ate, of the 
class of grains of size (area) a, n the number of incident 
light-quanta, again per unit area of plate, and k the number 
of grains affected J. If the finite cross section, a of the 
light-quantum is taken into account, a is to be replaced by 



'-['Va*- 



At first, the rapid increase of k with the size a as required 
by that formula seemed (qualitatively speaking) not only 
attained but even exceeded by the experimental results. 
This seemed to us to indicate that the sizes (areas) of the 
clumps of grains were under-estimated by us. In fact, for 

* Extensive experiments are being conducted in this laboratory to 
study the clumping of grains in different concentrations of the same 
emulsion. These investigations will be published later. 

t Slade and Higson, "Photochemical Investigations of the Photo- 
graphic Plate." Proc. Roy. Soc. xcviii. p. 154 (1920). 

% Silberstein, L., "Quantum Theory of Photographic Exposure,'' 
infra, p. 257. 



254 Messrs. Trivelli and Kighter on Silbersteins 

an estimate of the areas of all clumps one and the same 
average grain size (area) had been assumed throughout. 
Upon recalculating the results, however, and assigning the 
correct average grain size (area) to the single grains and to 
the different clumps, the very interesting fact was observed 
that the average grain size (area) increases from the single 
grains to the clumps of two, three, etc. The corrected results 
conform, even better than expected, to the above formula, 
with a finite a. 

Experimental. 

These experiments, although of an extremely tedious and 
trying nature, were performed with the utmost care and, to 
the best of our knowledge, all sources of error were either 
eliminated or reduced to a minimum. Only a brief descrip- 
tion of the experimental procedure will be given at this 
time, as a more extensive paper containing further experi- 
mental results is to be published in the near future with a 
detailed account of our methods of photomicrography, and 
in which all errors will be discussed fully. 

A simple silver bromide emulsion was used for these 
experiments having a speed of 112 and gamma 0*8 for six 
minutes development in an ordinary pyro-soda developer 
at 17° C. The average size of the grain is about 09yit 
diameter. 

The method of preparation of strips for sensitometric 
exposure is, briefly, as follows : — One 5 in. x 7 in. plate of the 
original emulsion is soaked in distilled water for one half-hour 
at 0° C. to 8° C. (All work for sensitometric exposure was 
done in a dark room by the aid of a dull red safelight, 
Wratten Series 2). The water is removed and a warm 
solution of gelatine, alcohol, and water is added and the 
whole solution heated in an oven for 20 minutes, while care 
is being taken not to heat over 10° C, because above this 
temperature much fogging takes place. With several such 
applications of the aforesaid solution the emulsion is entirely 
removed from the plate and the resulting solution made up 
to such a volume that it will give one layer of grains upon 
coating and drying. Some of the slides are used at once to 
get the clump frequency data. Those for exposure are 
backed with an opaque substance to prevent reflexions, then 
exposed in a sensitometer, developed to gamma infinity with 
a pyro-soda developer at 17° C, washed, and the developed 
silver removed with a dilute solution of chromic acid and 




Quantum Theory of Photographic Exposure. 255 

sulphuric acid. The strips thus obtained contain the unde- 
veloped grains, and by taking the difference the number of: 
developed grains is calculated. 

The data given below cover only the first or highest 
density step of a Hurter and Driffield sensitometric strip *. 
20 fields on each of 3 strips are employed to determine the 
developed grains. To determine the number of grains and 
clumps in the original one grain layer plate before sensi- 
tometric exposure and development, 10 fields on each of 
4 strips were used. By taking this large number of fields 
on several strips we obtained a much better average. The 
results in both the above cases are reduced to a number of 
grains or clumps per square centimetre of one grain layer 
plate. Then as the dilution is known, one may, with certain 
restrictions, refer back to the original plate. 

All photomicrographs were made at a magnification of 
2500 diameters and these negatives enlarged 4 times in 
printing. On the prints the grains and clumps are measured 
and counted, and then classified in class sizes (areas). The 
class sizes (areas) are to 0*2 /* 2 , 0'2 to 0*4 yu 2 , 0*4 to 0"6//, 2 , 
etc. The light source is a point source from a Pointolite 
lamp which is screened with a YVratten (H) blue filter to 
restrict the wave-length range and therefore increase the 
resolving power of the microscope. A cell containing 
copper sulphate solution absorbs heat rays and a cell con- 
taining a solution of quinine bisulphate excludes the ultra- 
violet light. The optical system is built up as follows : 
Cedar oil immersion condenser and objective, aplanatic 
condenser of numerical aperture 1*4, and Bausch and Lomb 
objective 1*9 mm. numerical aperture 1*3 in combination 
with a No. 6 compensating ocular. 

In the following is a table of our results. Column 1 
contains the number of grains in each clump, column 2 the 
average area of grains in corresponding clumps, column 3 
the number of grains times 10 -3 per square centimetre of 
original one-grain layer plate, column 4 the number of 
grains times 10 ~ 3 per square centimetre of developed one- 
grain layer plate, and column 5 gives the proportionate 

k 
number ^ of clumps affected. 

* The five remaining steps each corresponding to one-half the exposure 
of the preceding are now being counted and mapped out, and we hope 
to be able to publish the results obtained with them in the course of 
one or two months. 



256 Quantum Theory of Photographic Exposure. 



Number of 




N.10" 3 per 


£.]0~ 3 per 






2 


sq. cm. 1 grain 


sq. cm. 1 grain 


k 


in Clump. 


P • 


layer plate 


layer plate 


W 




(original). 


(developed). 




1 


0-75 4 


65770 


1086-0 


0*165 


2 


1-92 5 


1322-0 


594-0 


0-449 


3 


3-03 


664-0 


508-6 


0-766 


4 


4-88 1 


328-0 


2863 


0871 


5 


-r.^ 9 i unreliable 


249 9 
157-8 


240-0 
155-0 


0-960 
0-982 


6 


7 


8-6 


1052 


105-2 


1-000 


8 


9-8 
110 


526 

52-6 


52-6 

52-6 


1-000 
1-000 


9 


10 


120 


39-5 


39-3 


1000 


11 




26-3 
39-5 


26-3 
39-5 


1-000 
1-000 


'2 


13 




19-8 

19-8 

6-5 

13-2 

13-2 

132 

6-5 

3-3 


198 

19-8 

66 

132 

132 

6-5 

3-3 

G-6 


1-000 
1-000 
1000 
1-000 
1-000 
1-000 
1-000 
1-000 


14 


15 


16 


17 


18 


19 


20 


21 




6-6 
13-2 


13-2 
33 


1-000 
1-030 


22 


23 




3-3 


1-3 


1-000 


24 




1<3 


6-6 


1-000 


25 




66 


1-3 


1000 ■ 


26 




1-3 





1-000 


27 







33 


1-000 


28 




3-3 





1-000 


29 










1-000 


30 







1-3 


1-000 


31 




1-3 


1-3 


1-000 


32 


24-0 


1-3 


1-3 


1-000 


33 


25-0 


1-3 




1-000 



The agreement of the numbers of the last column with 
those calculated by Silberstein's formula (with N = 0*572 
and <r = 0*0973), as given in his paper, is manifestly a very 
pronounced one. The differences between the observed and 
calculated values are even in the case of the 3 grain clumps 
entirely within the limits of experimental error, particularly 
concerning the area measurements. 

In continuance of this work the theoretical formula will 
be subject to further tests, the results of which will be 
published shortly. 

Rochester, N.Y. 

January 24, 1922. 




[ 257 ] 



XIX. Quantum Theory of Photographic Exposure. 

By L. SlLBERSTEIN, Ph.D.* 

1. rriHE purpose of the present paper is to describe a first 
JL attempt at a light-quantum theory of photographic 
exposure, or of the production of the so-called latent image, 
with the immediate consequences of such a theory and some 
of its experimental tests. 

The silver-halide grains of an emulsion spread over a plate 
or a film base may be considered (apart from the smallest 
grains) as small flat plates, of comparatively small thickness, 
which in a dry emulsion lie almost parallel to the base. The 
sizes a (areas) of these plates range from submicroscopic ones 
up to 18 or 20 square microns. Even the most uniform 
emulsions obtainable in practice consist of grains of different 
sizes, the distribution of sizes among the grains being in 
each case characterized by what is technically called " the 
frequency curve " of an emulsion. In what follows the 
number, per unit area of the photographic plate, of grains 
whose areas range from a to a -{-da will be denoted hyf(a)da 9 
and a photographic emulsion will be shortly referred to as 
being of the type /(a). For certain emulsions /(a) is, with 
good approximation, an exponential, for others a Gaussian 
error-function of the area a, and so on. 

2. Without, for the present, dwelling any longer upon 
details of this kind, we may pass at once to our main 
subject. 

According to Einstein's well-known hypothesis of 1905 
light does not consist in a continuous distribution of energy, as 
in the classical theory, but is entirety split up into light quanta 
or discrete parcels of very concentrated monochromatic light, 
each parcel containing a quantum of energy, hv = h\/c, 
in obvious symbols. Somewhat more generally we may 
assume that only a fraction olE of the total light energy 
E is thus split into concentrated parcels, the remainder 
E = (l — ct)E being distributed continuously t> without 
however prejudicing the possibility of E being zero. 
Then, if E be the energetic value of a monochromatic 

* Paper read December 28, 1921, at the Toronto meetiug of the 
American Physical Society in affiliation with Section B of the American 
Association for the Advancement of Science. Communication No. 139 
from the Research Laboratory of the Eastman Kodak Company. 

t Somewhat as in E. Marx's theory of " concentration places " or of 
" light specks'' as suggested by Sir J. J. Thomson. 

Phil Mag. S. 6. Vol. 44. No. 259. July 1922. S 



258 Dr. L. Silberstein on a Quantum 

" exposure " of wave-length X, the number of light-quanta 
contained in it will be 

n= Tc X (1) 

From a recent conversation with Einstein, there are 
weighty reasons for making a=l. But since we do not 
prejudice its value, there is no harm in retaining this 
coefficient in the formula. 

Now, let us assume that the necessary and sufficient con- 
dition for a silver-halide grain to be affected, i. e., to be made 
developable (entirely or in part) is that it should absorb one 
light-quantum. 

Moreover, let us assume that a grain * does absorb a light 
quantum whenever it is fully hit by one, of a sufficiently 
high frequency v c or of a wave-length not exceeding a 
certain value X c . 

There are perhaps some experimental hints or more or less good 
reasons for making these two assumptions, but we need not stop to 
consider them here. It will be time to reject or to modify them when 
they are contradicted by photographic experiments. Nor is it necessary 
to enter here upon the mechanism by means of which a silver-halide 
grain is affected by a light-quantum, whether it be the knocking out of 
an electron, as suggested by Joly, or something entirely different. For 
none of such details will influence our main argument, to be treated in 
the next section. Only when we come to consider the dependence of the 
photographic effect upon the wave-length will it be interesting to con- 
sider the photoelectric hypothesis and necessary to take account of the 
fact that a photo-electron is not liberated unless the frequency exceeds 
a certain, the so-called critical value. Under these circumstances 
A e appearing in our second assumption will stand for the critical 
wave-length as known from Photo-Electricity. 

Again, whether a grain being " affected " is made developable in part 
only or throughout its whole area (no matter how large) is, in view of 
the* kind of the contemplated experimental tests, of no great importance. 
As a matter of fact, however, there is good evidence that a grain is 
always made developable as a whole, no matter what its size, and this 
seems even to hold for " clumps " or aggregates of several smaller 
grains, as will be explained hereafter. If so, then our formulae, to be 
"developed presently, will give not merely the number of affected grains 
but also, by integration, the total " mass " made developable and hence 
also the photographic density. But we may as well remain content with 
the formulae for the number (~k) of grains affected, and count these in all 
experimental tests. This, far from being a disadvantage, will enable us 
to subject the proposed theory to more precise, though at the same time 
more severe tests. 

One more remark. It will be understood that when we come to adopt 
the photo-electric hypothesis, a grain " affected " will stand for a grain 

* Or perhaps, more generally, one of every p grains hit, where p 
is a number to be determined by experiment, but presumably equal 
to unity. 




Theory of Photographic h\vposure. 259 

which as a whole has been deprived of even a single electron only. 
It need not lose more than one electron in order to be made entirely 
developable. According to Professor Joly's original hypothesis * the 
latent image " is built up of ionised atoms or molecules." In our con- 
nexion, this does not mean that for every pair of atoms, say Ag Br, 
there is one electron liberated. Since every grain of silver bromide 
(as well as of silver chloride) is a crystalline, to wit a simple cubic 
space-lattice arrangement of Ag and Br atoms t, we may as well con- 
sider the whole grain as a single molecule. Such a crystalline structure 
being hit by a light-quantum and deprived of but a single electron, 
may well become susceptible throughout to the subsequent action of 
a developer. 

3. With the assumptions just made the question is reduced 
to a mere problem in probabilities. 

Consider first the ideal case of equal grains. Let there 
be upon an area S of the photographic plate (feay unit area) 
N grains. Let a be the size (area) of each of them divided 
by S, and let n light-quanta impinge upon S, due allowance 
having been made for those which may be reflected or 
absorbed by the gelatine. The problem consists in finding 
the number k of grains hit and (if p = l) affected by this 
light exposure. 

Roughly, this simple problem can be treated as if -A 7 , k 
were continuous quantities, in the following way, familiar 
from many other instances (" mass-law "). At an}^ stage 
the number of unaffected grains is jST—k, representing an 
available fraction a(N — k) of the total area S. Thus, if 
further dn light-quanta be thrown upon S, and if their trans- 
versal dimensions be negligible as compared with those of 
the grains %, the corresponding increment of lc will be 

dk = a(]S r — k) dn } 

and since k = for ?i = 0, this gives at once 

k=B(l-e- na ), (A) 

which in fact will presently appear to be correct enough 
except for the practically unimportant cases of small iV or 
small JS r —k. 

More rigorously, but provided always that JV at least is a 
large number §, the required formula can be obtained by the 

* Nature, 1905, p. 308. 

t R. B. Wilsey, Phil. Mag. xlii. p. 262 (1921). 

% An assumption which will be given up in the sequel. 

§ 'Which will be the case if S is taken large enough. Since plates 
and films in actual use contain as many as ]0 r) grains per cm.' 2 , S can be 
made as small as one-hundredth mm. 2 , and even less. 

S2 



260 Dr. L. Silberstein on a Quantum 

following reasoning. The total area of silver halide being 
a fraction Na of the area S of the plate, each of the nJSa 
light-quanta will fall upon some grain. Of these nNa 

quanta one, say the first, will hit one grain, the next ^ 

N . 
quanta will fall upon another grain, the next _ will hit 

yet another, and so on, up to ^ 7 _ , .. quanta for the Irth 

grain *, Thus the required relation between k and n 
will be 

_1 1 1 l_ 

an ~ JX + N-l "" + N-k + 2 + N-k + r 

Now by a well-known theorem of Analysis 

J + i + l+ •••• +-=0 + logm + e(m), . . (2) 
JL o nx 

where C is Euler's constant and 0<e(«i) < 1/m. Thus in 
our case 

lo gyZTk= an + £ 

where f lies between — 1/A 7 and l/(J¥—k) or practically, 
since iVis at any rate a large number, 

e<?= £ (ir-*)<^. 

Whence, 

fc=JV(l— *-**-£) (3) 

In practicable experimental tests (counts of affected grains 
of a given, narrow size-class) the role of the correction f 
whose value f=e(iV r — k) can at any time be found by (2), 
may become perceptible only when the contemplated grain 
class is near exhaustion. 

Thus, apart from such extreme cases, we have again, 
as in (A), the simple formula 

k = A\l-e- na ), (4) 

which, though approximate only, will turn out to be 
accurate enough even for moderate values of N. 

A thoroughly rigorous treatment of the probability 
problem valid for any numbers iV, ?i, seems to be the 

* It is scarcely necessary to say that statements such as U N/N— 1 
quanta hit another grain " are to be taken statistically as relating to 
averages over a group of many trials. 




Theory of Photographic Exposure. 201 

following one. Of the n quanta the total number falling 
each upon some of the N grains will be * 

m = Na n, 

JS r a being the total area of silver halide, with S as unit area. 
Thus the problem is reduced to finding the distribution of m 
quanta among JS 7 grains. Now, let p m (i) be the probability 
of affecting, in a single trial, a number i of the JV grains, by 
the m quanta. By a combinatorial discussion which may 
be omitted here, I find 

P»W = jy»(]y-i)V ( } 

where a mi may be most shortly described as identical with 
the number of ways in which a product of m different 
primes can be decomposed into i factors. These numbers, 
which will be known to many readers from combinatorial 
algebra, have the obvious properties 

a m i=a mm —l, for any m, 

a mi = for i > m, 
and satisfy the general recurrency formula 

which enables us to write down successively without trouble 
any number of them. 

Thus, up to wi = 10, we have the following- table which the reader 
may continue to extend at his leisure. Columns correspond to con- 
stant m, and rows to constant i. 

1111111 1 1 1 

1 3 7 15 31 63 127 255 5 LI 

1 6 25 90 301 966 3025 9330 

1 10 65 350 1701 7770 34105 

1 15 140 2401 13706 76300 

1 21 266 3997 37688 

1 28 462 7231 

1 36 750 

1 45 

1 

We may mention in passing that any a m i can be represented by f 

«»••= n D'"'- (i) {i ~ 1)m+ (») ('- 2 >*"- ± (!)]■ 

But for any numerical applications the table will be found more 
convenient. 

* In a large number of trials of the same experiment. 
t Cf. e. g. Lehrbuch d. Kombinatorik by E. Netto, Leipsig, 1901, 
p. 170. 



262 Dr. L. Silberstein on a Quantum 

By formula (5) we have for instance the probability of 
hitting but one grain 

Pm\ Ij = jy^-l -> 

which is an obvious result. The probability of hitting two 
grains will be 

i?m (2) = (2— i-l)(j\r-l) . jy*- 1 , 

which is (approximately) 2 m N times as large as the preceding 
probability, and so on. If m be kept constant, p m (i) will 
increase with growing i up to its largest value for i = m, 
if ra<iV, or for i — N, if m > N. But it would be futile to 
expect, on an average, that distribution to which corresponds 
the greatest probability *. For all other distributions have 
some generally non-negligible probabilities and these are by 
no means symmetrically spaced with respect to the largest 
one. The only reasonable way of determining the number k 
is to define it as the average of i taken over a large number 
M of trials. Out of these M trials a number M p m (i) t of 
trials will give each i grains affected, and the total number 
of grains affected in all M trials will be XM^p m (^), to be 
summed from i = l to i = m, if m-<N and to i — N\i m >iV r ; 
but since a m i = for i > m, we can as well extend the sum 
in each case from i=l to i = N. 

Dividing this sum by M we shall have the average number 
of grains hit in one trial, i. e., by (5) 

7Vf N in - 

This, with m — Nan, is the required rigorous formula for 
the number of grains affected, i. e., hit once or more. In 
order to see how this complicated formula degenerates into 
(4), which, of course, will be our working formula, develop 
the sum in (6). Collecting the terms in i^ -1 , N~ 2 etc., and 
taking- account of the values of a m ,\ it will be found that 



l mij 



, 1 (m\ 1 fm\ ( 1 V" -1 

* If, say, ?n<:N, the most probable distribution is the equipartition 

(a quantum per grain), corresponding- to p m (m)= T^ m /j^^_ \ t ■ This 

would give as the number of grains affected k = m = Nna, or just the 
first term in the series development of (4), which would be hopelessly 
wrong- unless mN were very small. 

t "With a deviation ruled by Bernoulli's law. 



Theory of Photographic Exposure. 263 

or, dividing by N and subtracting from unity, 

'-i-»(t)(-»)*-*o(-ir- 



N 
i. e. ultimately 

-»-(»-ir m 

This is rigorously equivalent to or identical with (6) for any 
m and X Now, for large JN", and any m, equation (6') in 
which (1 — l/N)*=:l/e, asymptotically, gives at once 

k -™ 

-r T =l — e * = l — e^ va . 

This is the connexion between the rigorous arithmetical 
formula and the exponential one. 

It is needless to insist that under the conditions prevailing 
in all practicable experimental cases there is more than suffi- 
cient mathematical accuracy in formula (4) 

k = N{l~e- na ) J 

which, apart from minor modifications, will henceforth be 
used in what follows. 

This formula is of the familiar type proposed (1893) by 
Elder, with the notable difference, however, that while his 
exponent contained a free " parameter " or coefficient to be 
evaluated empirically and principally depending upon "grain- 
sensitivity " and wave-length, both coefficients in (4) are 
completely determined, and the exponent moreover shows 
an explicit and most essential dependence upon the size (a) 
of the grain, and in the right sense too, i. e., giving an 
increase of the " speed " with grain size. The comparison 
with experimental facts of Elder's and of a number of other 
formulae, constructed empirically, is too well-known to be 
discussed here * Suffice it to say that, although it repre- 
sents to a certain extent the photographic behaviour (the 
" characteristic " curve) of some emulsions, and particularly 
those with what is termed an " extended toe," it cer- 
tainly shows considerable deviations from the observed 
characteristic curves. Yet it will not be forgotten that all 
these comparisons bore upon the resultant total photographic 
densities, containing or integrating the effects of grains 
belonging to a broad range of sizes (a), instead of equal 

* Cf. for instance, a paper bv Dr. F. E. Ross, Jonrn. Opt. Soc. Amer. 
vol. iv. p. 255 (1920). 



264 Dr. L. Silberstein on a Quantum 

grains, so that no better agreement could be expected. 
The refined experimental tests which are now in progress 
in this laboratory, and by means of which it is hoped to 
corroborate the proposed theory, deal, as they should, with 
separate size-classes of grains. 

But questions concerning the comparison of the theory 
with experiment will be treated in a later part of the present 
and in subsequent papers. 

4. Passing next to the case of an emulsion of any type 
/(a), it can be easily proved that the approximate formula 
(4) will hold for each class of grains separately. In order 
to see this it is enough to consider the case of two distinct 
classes of grains. Thus, let there be N± grains of size a l 
and N 2 grains of size a 2 spread over the (unit) area S of the 
plate, and let n light-quanta be thrown upon S. Of these a 
number m 1 = niV r 1 a 1 will fall on the a r grains and a number 
m 2 = nN 2 a 2 upon the a 2 -grains. It remains only to be found 
how many ax-grains will be hit by the m 1 quanta, and how 
many of the a 2 -grains will be hit by the m 2 quanta. Now 
each of these is a problem of the kind we have already 
treated. The number k x of ax-grains hit will be given by 

1 J_ 1 

i^x + iVi-l^ + iV 1 -/r 1 + l- naiJ 

and similarly for the a 2 -grains, so that k x and k 2 will each be 
determined by the previous formula for k with J\ 7 , a replaced 
by JSr iy ai, and N 2% a 2 respectively. Similarly for an emulsion 
consisting of three or more classes of grains. 

Thus, also, for an emulsion of any type /(a), the number 
of grains ranging from a to a-\-da hit and (if £>=1) affected 
by n impinging light-quanta will be k a da, where, apart from 
the correction f 

The total area * of silver halide affected or made developable 
will be found by extending the integral 



K- 



\akada (8) 



over the whole range of sizes, say from a 1 to a 2 . 

If, for instance, f(a) = Ce~ fl - a , say from a = a 1 to a 2 , where 
C, ix are constants, as in the case of some films and plates 

* It will he kept in mind that a stands for the " efficient " area of a 
grain (plate), i.e. for the orthogonal projection of the grain upon the 
film base. 



Theory of Photographic Exposure. 2G5 

investigated in this laboratory for their frequency curves, 
then, with A written for the total area of silver halide, 

But this only by way of illustration. The fundamentally 
important thing will be formula (7), applicable to each 
size-class of grains separately. .In fact the experimental 
verification of the theory now in progress in this laboratory 
deals, not so much with K but, as it should, microscopically, 
with k = k a for each class of grains separately, including 
clumps of grains. 

Before passing to a further discussion and development of 
the elementary formula (7), but one more remark concerning 
the presence of more than one layer of grains. The case of 
two or more layers will at once be seen to be reducible to 
that of a single layer. In fact, either a grain of, say, the 
second layer and of size a is not shielded by any of the first 
layer or else it is thus screened off and only a part b of it 
remains uncovered. In the former case the grain in question 
will simply be classified among those of size a of the first 
layer, and in the latter case among those of size b. This will 
hold with respect to the exposure to the impinging light- 
quanta, and b will also be the contribution of the grain in 
question to the photographic density ; for its covered part 
will remain inoperative. Similarly for three or more layers. 
In fine, the presence of a plurality of layers of grains will 
modify only the frequency curve N a ~f{a) which would 
otherwise belong to a single layer. We shall henceforth 
assume that this factor has already been taken into account 
in constructing the function /(a) or in microscopic counts of 
the grains within every particular size-class. We disregard 
here, of course, such factors as a possible absorption of light 
in additional strata of gelatine. 

5. Dependence upon wave-length. — Once more return to 
the elementary formula (7) or (4). Denote by s the ex- 
ponent so that 

N L e ' 

Under the more or less implicit assumption that the trans- 
versal dimensions of a light-quantum are negligible in 
comparison to those of a grain, we had s = na. But it will 



266 Dr. L. Silberstein on a Quantum 

be seen presently that such an assumption is too narrow and 
unnecessarily so. 

In fact, substituting the number n of light-quanta from 

equation (1), the exponent s* will become s = y-i?X, or, if 
we put for brevity 

f3 = ot/hc, . (9) 

which may be considered as a constant, 

s=/3E.a\ (10) 

Here E is the incident light energy (exposure) and X the 
wave-length of the light assumed to be monochromatic. 
Thus the sensitivity exponent would be directly proportional 
to the wave-length, and the number k would, for constant 
E and a, increase steadily with the wave-length of the 
incident light up to the photoelectric critical value X c and 
then drop suddenly to zero, 

k 

j r = l-e- const -\\<\ n 

k =0, X > X c . 

Now, such a sensitivity curve does not seem to resemble the 
familiar experimental sensitivity curves which show a more 
or less gentle maximum followed by a gradual decrease 
down to zero. It is true that such experimental curves | 
represent the resultant effect due to grains of a whole range 
of sizes, so that the k — X curve belonging to a single class 
of (equal) grains may well be of the said abrupt type, — a 
question to be decided only by micro-spectrographic experi- 
ments and counts now in progress. Yet it seems advisable 
even at this stage to provide for the possibility of smooth 
maxima preceding the critical wave-length X c . 

This might be obtained by attributing to grains of different 
sizes different values of X c . For then, although the curve 
of each grain class would end abruptly , the superposition of 
such curves ending over a range of different abscissae, might 
properly displace and smooth out the resultant maximum. 
The correctness of such an assumption (X c a function of a) 
can at any rate be tested by direct experiments %. 

* s/JE can be referred to as the u sensitivity exponent." 
t Apart from the fact that they are not taken for E= const. 
\ Preparations for such experiments are now being made in this 
laboratory. 



Theory of Photographic Exposure. 2G7 

Another way is to take account of the possibly /im'^ trans- 
versal dimensions of: the light parcels, which may perhaps be 
comparable with those of the lesser silver halide grains. 

Let us assume, therefore, that a light-quantum, of suffi- 
ciently high frequency, becomes efficient in affecting a grain 
only if it strikes it fully, or almost so. To fix the ideas, let 
r be the equivalent radius of a grain, i. e., such that 



and similarly let p be the average radius of the transversal 
section of a light parcel (so to call the space occupied by a 
quantum of energy). Then the efficient area of a grain to be 
substituted instead of a, will be 

a' = 7r(r — p, 2 ~a 1— ^ , 

and we shall have for the exponent s, instead of (10), 

s = na'=/3Fa [l-^Tx, for p<r< \< : \„. (10a) 

and 5 = for p > r or X > X c . 

It remains to assume in a general way that p, which may 
be the average of section-radii different even for parcels of 
the same wave-length, is itself a function of the wave-length 
increasing with X, without prejudicing, however, the parti- 
cular form of this function. Certain easily ascertainable broad 
features of such a function and thence also of the resulting 
factor in s, 



£(\)=\[ 



(ii) 



will suffice to ensure a maximum of the sensitivity exponent 
between X = and X = X e . The value of X c itself may still 
turn out to depend on the size a of the grain and on its 
physical conditions as well. Every process which will make 
the liberation of a photoelectron from the grain (crystal) 
easier will lengthen X c . Part of the effect of sensitizing 
may arise in this manner. But questions of this kind must 
necessarily be postponed until some further experimental 
data are gathered. Of such a kind is also the question 
whether p (which, for a given X, may also extend over a 
whole range of values) attains at all the semi-diameter r of 
even the smallest of the actual grains, and whether the 
corresponding wave-length A entailing the vanishing of s, 
exceeds or is smaller than X c as derived from direct photo- 
electric experiments. In absence of all knowledge concerning 



268 Dr. L. Silberstein on a Quantum 

the spatial properties of light-quanta it would be utterly 
unjustified either to deny or to assert that their lateral 
dimensions are at all comparable with those of a silver 
halide grain (of the order of one-tenth up to several 
microns) *. If, by way of illustration only, p is propor- 
tional to a power of \, say p = b\ K , the only condition for the 
existence of a maximum of $(X), and therefore of sensitivity, 
will easily be found to be k>0. If this be satisfied, the 
maximum will occur at a wave-length \ m given by 

increasing with the diameter of the grain and bearing to X 
the fixed ratio 

^=(2*;-r-iy/\ 

As a matter of fact, the maximum sensitivity is known to shift 
(by two or three hundred A.U.) towards the red by making 
the grain coarser. But thus far too little is known of the 
quantitative aspect of such an effect to entitle one to con- 
sider the above equation as anything more than an illustrative 
example. The precise form of the function p = p(\) can only 
be derived from spectrographs experiments followed by 
microscopic grain counts, or if arrived at by a guess, has 
to be verified by them. Such experiments are now in 
preparation in this laboratory, and their results will be 
reported in due time. A shift of the maximum sensitivity 
towards the red or the infra-red can, of course, be brought 
about by a function form more general than a mere positive 
power of the wave-length. 

6. Generalities, and 'preliminary account of experimental 
tests. — The chief and most immediate consequence of the 
proposed theory is the essential dependence of the propor- 
tionate number of grains affected, k/J¥, on the size a of the 
grain, viz., the rapid increase of the former and, therefore, 
of " the speed " of an emulsion with the latter. Now, it has 
been known for a long time that (cceteris paribus) the speed 
increases notably with the size of the grain, and we shall 
see from the experiments to be described presently to how 

* According to E. Marx, Annalen der Physih, xli. pp. 161-190 (1913), 
the volume of a light-parcel, which according to him is only a u concen- 
tration place " within a continuous distribution of energy, is proportional 
to X 4 and amounts for D-light to almost 8.10 -7 cm. 3 , which even with 
a length of 10 cm. (200,000 D- waves) would still give a section area 
8.10" s cm. 2 , just of the order of about the average grain area. There is, 
of course, nothing cogent about Marx's estimate, yet the matter is not 
without interest. 



Theory of Photographic Exposure. 269 

large an extent this is actually the case. But perhaps 
the most tangible proof of the essential correctness of the 
assumption of spatially discrete as against continuous action*, 
seems to be the mere fact, disclosed by microscopic counts, 
that out of a number of apparently equal grains subjected 
to a sufficiently weak exposure one or two are affected while 
the others, nay their next neighbours, remain perfectly 
intact. It would be in vain to ascribe to these survivors a 
greater immunity or indifference to light. For it is enough 
to protract the exposure a little to make them succumb in 
their turn. Now such a behaviour is most typical of rain 
as contrasted with flood action, and the discrete light-quanta, 
hitting now this and now that grain, appear to be a most 
natural inference, while all attempts to bring into play the 
individual ''sensitiveness" of the units seem to involve 
considerable difficulties. 

As to the dependence of: the number of grains affected 
upon the wave-length, little more of interest in the present 
connexion is known than the qualitative fact of a shift 
towards the red of the maximum sensitivity with increasing 
size of the grain. Moreover, the available curves repre- 
senting the sensitivity across the spectrum concern the 
emulsion as a whole and not the separate a- classes of grain 
with which we are primarily concerned. Spectrographs 
and spectrophotometry experiments of such a kind, to be 
aided by direct photo-electric measurements, are now in 
progress in this laboratory, and all discussions involving 
wave-length will best be postponed until the results of 
these experiments and of laborious microscopic counts are 
forthcoming. 

Before passing to the mentioned quantitative test of the 
dependence on size, but one more general remark. The 
reader will have noticed the complete absence of the time- 
variable in all our formulae, the exposure entering only 
through the total number n of light-quanta or through the 
energy E which, in obvious symbols, is ^Idt. The pro- 
posed theory, therefore, as thus far developed, does not take 
any account of the little infringements against the reciprocity 
law t, iu short, of the so-called " failure of the reciprocity 
law." Now, it is by no means my intention to deny the 

* A rain as against a flood, of light, that is. 

t This early law asserted the dependence of the photographic effect 
(density) upon I and t only through the total incident energy or ex- 
posure \ Idt. For constant intensity this is It, whence the name of the 
law, relating to intensity and exposure-time as factors of a constant 
product. 



270 Dr. L. Silberstein on a Quantum 

reality of these infringements which have been extensively 
studied and condensed into empirical formulae by Abney, 
SchwarzschiJd, Kron and others. But it has seemed inad- 
visable to encumber the very beginnings of the proposed 
theory by complicated details of such a kind*. 

The failure of the reciprocity law can more profitably be 
taken up later on, after the fundamentals of the theory have 
been somewhat solidified and extensively tested, and the 
prospects of mastering the "failure" theoretically are by 
no means averse, a very promising scheme seeming to lie 
in the possibility (suggested by Joly and taken up by 
H. S. Allen) of the liberated photo-electrons being regained 
by some of the grains which were deprived of them by 
previous impacts of light-quanta. In fine, the failure of the 
reciprocity law as well as the facts known under the head of 
" reversal " have at first to be neglected and considered 
as future problems for the light- quantum theory united with 
Joly's photo-electric theory, problems to which these com- 
bined theories seem well equal. 

To pass to numerical facts, a short description will now be 
given of the results of certain experiments undertaken in 
this laboratory by A. P. H. Trivelli and Lester Righter f 
which seem to corroborate the proposed theory most 
emphatically. In order to have a much wider range of 
sizes a than is usually afforded by the single grains, 
Trivelli and Righter applied their counts and area measure- 
ments to clumps of from one to as many as 33 grains, 
basing themselves upon the well- supported assumption that 
if one of the component grains be affected, the whole clump 
is made developable. (This, at anj r rate, is the behaviour 

* R. E. Slade and G. I. Higson, " Photochemical Investigations of the 
Photographic Plate," Proc. Roy. Soc. xcviii. pp. 154-170 (1920), on the 
contrary, make the failure of the reciprocity law their point of departure. 
They mention at the very beginning (p. 156) the possibility of a light- 
quantum theory and write the Elder-type of formula in I, t, remarking 
even that its coefficient would have a different value for each size of 
grain, but being discouraged by or rather preoccupied with the failure 
of the reciprocity law do not enter into the details of the probability 
problem, which would have disclosed them the structure of that co- 
efficient, and without much ado dismiss the quantum theory as 
'* impossible." Independently of Slade and Higson the possibility of 
a discrete theory (radiation in "filaments ") is mentioned by F. E. Ross, 
Astrophys. Journ. vol. Hi. p. 95 (1920). Dr. Ross, without being 
prejudiced against such a theory, notes even that it would lead ration- 
ally to a mass-law, but does not enter into the details of the probability 
problem and does not develop the theory. 

t For technical derails of these laborious experiments, see Trivelli and 
Righter's own note in this issue of the Phil. Mag. p. 252. 



Theory of Photographic Exposure. 271 

of the larger, flat grains piled upon each other in part, 
although the smaller, spherical grains, in less intimate 
contact, may perhaps behave differently.) Such being the 
case, their experimental results should be covered by our 
formula with a written for the area of the whole clump, no 
matter how large and how numerous its components. This 
has seemed a rather severe test but the more so tempting and 
instructive. Since all the classes of clumps were given, in 
each trial, a unique exposure (through a blue filter specified 
toe. cit.) and there was no question of varying X, it will be 
most convenient to retain in the corresponding formula the 
original light- quantum number n as the parameter common 
to all clumps. Thus the formula to be tested becomes 



— =1 — e , s=na =na 



(»-»)'■ 



or somewhat more conveniently for computations, if o~ = 7rp 2 
be the (average) area of the transversal section of a light 
parcel, 

lo g (l-4) = -™[l-^]\ . . (12) 

In the following table the first column gives the number 
of grains in a clump, the second the average area a of a 
clump in square microns, and the third column the per- 
centage of clumps affected out of all (i\ 7 ) clumps of each 
kind originally present, i.e. 

100 k 

as deduced by Trivelli and Righter from their observations. 

Clumps of ain^. y oba> y c ^ Ay. 

1 grain O-75-i 16-5 162 +0-3 

2 grains 1-925 44-9 48'4 -3"5 

3 , 3-03 76-6 68-9 +8-3 

4 4-88 87-i 87-3 -0-2 

5 „ Crls 96-" 933 2-7 

6 ., 7-4-2 98-2 964 1-s 

7 „ (8-6) 100 98-0 2-o 

8 , (9-8) 100 99-6 0-4 

9 , (11- ) 100 99-8 0-2 

10 „ (12- ) 100 100 0-0 

etc., etc. etc., etc. idem. idem. idem. 

32 grains >24 100 100 0-0 

33 „ >25 100 100 00 

The most reliable a-values are those for the clumps of one 
and of two grains, being averages of the largest numbers 



272 Dr. L. Silberstein on a Quantum 

of individual clumps ; the following a's are gradually less 
reliable ; from the 7-grain clumps onward the areas 
(bracketed) are only extrapolated, but since here y has 
practically reached 100, no greater accuracy is needed. 
The fourth column contains the theoretical values of y 
following from formula (12) with the constants, determined 
from observations 1, 2, 4, 

n = 0*572* per a 2 , l 

„=O097^, I ■'•■<«•> 

or p — 0*176//-. The fifth column gives A?/=y ob8 . — z/ ca i c> . 
The agreement is certainly very pronounced, the differences 
being, perhaps with the exception of the third, well within 
the limits of experimental error chiefly in the a-estimates. 
The fitting could be made even closer by retouching the 
constants n, a, but this is scarcely worth while, the formula 
itself being of a statistical nature, and the agreement being 
good enough as it stands. The same is manifest also from 
the figure giving a graphical representation of the last 
columns of the table. 




_fl - a oaoo — e — o — o o - o — o — ooo g ooooo c — e- 



— CALCULATED 
• OBSERVED 



10 15 20 £5 



The reader might think that the finite section cr, or 
whatever this parameter may stand for, has been forced 
upon the light-quantum and that tho observations might 
perhaps be as well represented with cr = 0, and another value 
of n. But actually, just the contrary has been the case, 
inasmuch as the author first tried the simpler formula 
log (l — k/fil)=—na, and then only found himself compelled 
to take in the correction factor as given in (12). In fact, 



Theory of Photographic Exposure. 273 

dividing the observed values of log (1 — kjN) by the areas 
the reader will find that the quotient increases considerably 
and systematically, apart from a casual drop at the fourth 
clump, throughout the whole series of the clumps. Thus, 
the correction factor seems to come in quite spontaneously. 

On the other hand, there is nothing unlikely about the 
values of either of the constants (12 a). Our units of area 
being here square microns, we should have 57 millions light- 
quanta per cm. 2 (about which judgment has to be suspended 
until absolute energy measurements are available), and as 
the cross section of the space occupied by each of them 
(on an average) a little less than one-tenth of a square 
micron or a diameter of about 0*35 micron. Since each is 
presumably of about the order of a million wave-length 
long, they are rather slender at that cross section, and, 
instead of light parcels, as they were called above, would 
perhaps deserve rather the name of light darts. In Einstein's 
own theory there is nothing on which to base an estimate of 
the volume occupied by a light-quantum, but on Marx's 
less radical views this is about 8.10" 7 cm. 3 for D-light and 
proportional to \ 4 , and therefore in our case (narrow blue 
spectrum region with maximum at A=0'470yLt) about 
3.10 -7 cm. 3 , which with the said cross section would mean 
a length of 3.10 6 //. or over six million wave-lengths. But 
this by the way only. The important thing is to see whether 
the above numerical value of the average cross section of 
blue light darts will continue to fulfil its function with regard 
to the remaining "steps" (weaker and weaker exposures) 
of plates coated with the same emulsion, the above being the 
highest " step." These have just been completed in this 
laboratory, ceteris paribus with the above one, and are now 
being subjected to counts and area measurements. This 
material will also serve to test the constancy of k/ N if, 
varying n and a', their product is kept constant. 

An account of the results of these and of several other 
experiments now in progress will be given in future papers. 

I gladly take this opportunity to express my best thanks 
to Dr. C. E. K. Mees for having proposed to me the problem 
of " discriminating, if possible, between the consequences of 
a discrete and a continuous exposure theory/'' and to my col- 
leagues Trivelli and Righter for furnishing the results of 
their experiments. 

Rochester, N.Y.. 

January 19, 1922. 

Phil. Mag. S. 6. Vol. 44. No. 259. July 1922. T 



L 274 ] 



XX. An Analytical Discrimination of Elastic Stresses in an 
Isotropic Body. By R. F. Gwyther, M.A* 

Sir G. B. Airy obtained from mechanical considerations 
(British Association Reports, Cambridge, 1862) a solu- 
tion in Cartesian coordinates of the mechanical stress- 
equations, but he ignored all elastic requirements. In one 
sense this paper may be regarded as an extension of Airy's 
scheme, though it has nothing in common with that scheme 
either in general plan or detail. 

The method is purely analytical, depending upon general 
solutions of the mechanical stress-equations and upon the 
development of a scheme for the selection of those stress- 
systems which satisfy the stress-strain relations, briefly 
called Hooke's law, from the general mechanical stress- 
systems. 

It is shown that the elements of a mechanical stress 
depend upon an arbitrary primary stress-system, and, to 
form a connexion with the stress-strain relations, I introduce 
a subsidiary, but allied, stress-system which is such that the 
vector system naturally deduced from it possesses inherent 
qualities distinctive of the displacement corresponding to an 
elastic stress-system. 

The main body of the paper consists in discussing the 
requirements necessary to ensure that the stress-system 
should be an elastic stress-system. 

The displacement, according to this method, becomes 
somewhat incidental, however necessary, and the elements 
of stress are given prominence. There are no displacement 
equations. 

In the first instance I deal with a body under tractions 
only, and extend the scope of the results later. 



I. Introduction. 

1. By first treating certain ancillary matters as lemmas, 
the steps in the final stages can progress more 
steadily. 

The mechanical equations of stress in a body under 
* Communicated by the Author. 



Analytical Discrimination of Elastic Stresses. 275 
tractions only are, in Cartesian coordinates, 

"d.v B*/ "dz 
3.1* By B~ 

B* By + d~^ "' W 

and they are identically satisfied by the values 

B- 2 By^ByB*' 

B^l 

3. 



Q = 



R_ ^.cW, 

*"" V 3*" 



T = 
U = 



B^'B- 



b 2 3 , 2 avs 

B* 2 B#B* 






3#By' 


B^h _ j^h 
d« a B«By 


B 2 ^ 3 


b 2 ^ av, 

B^B* B3/ 2 


B 2 f 3 
3y3-~' 


B 2 # 3 BVi B 2 ^ 2 
B#B# B#B# ByB^ 


• • • (2) 



These contain six arbitrary and general functions, and 
form the general solution of (1). 

Lemma A. 

These six arbitrary functions have the same mode of 
resolution on transformation of axes as the elements of 
stress. 

For proof I use the method employed in the subject 
of differential-invariants. 

Imagine the axes of coordinates to be rotated about their 
own positions by the infinitesimal amounts co z , e» y , o) z , and 
consider the consequent changes in whatever quantities 
we may be considering — components of a vector, elements 
of a stress, etc. These changes will be linear functions of 

60 x , ftty, (O z . 

T2 



276 Mr. R. F. Grwyther on an Analytical Discrimination 

For example, in the case of the components of a vector 
u, v; iv, the changes will be : 

in u, —(D y w-\-co z v, 
in v, —cd z u + w x w, 
in w, —co x v-\-oc>yU. 

These changes will now be represented by partial dif- 
ferential-operators L\, 2 , Xl 3 arranged to produce the 
coefficients of co x , w v , co z in the quantities considered 
respectively. 
. Thus, for vectors generally, 

o 3 a 

This covers direction-cosines (Z, m, n), and may be made 
the basis of most operators. For example, we may deduce 

for stresses generally : 

ni = 2S (^-li) +(R - Q) l- u 9 V T ^ 

The differential coefficients A A A resolve as corn- 
er o t V o^ 
ponents of a vector, but, for simplicity, we must introduce a 
different notation. 

Write d x for B/B«, c? 2 for 3%, tf 3 for d/3*, and df„ for 
d 2 fdx 2 , d 12 for B 2 /d<^, d 22 for B 2 /3/ 7 and so on ; then 
for first differential coefficients 




of Elastic Stresses in an Isotropic Body. 211 

as for vectors, and we deduce for second differential co- 
efficients 

Ql = , ' 1, M ! "'' l 'Bl', +2 '' !i trs)" (4 " 4, S' 
n$ = ^mt^ 3 ^hz + u ™ farr w~ (**-*«) §^ 2 - 

It is required to establish that in (2) 1? 6 2 , 3 , yfr lt i/r 2 , yjr 3 
act as stresses. Actually I shall assume that the operators 
n i? Xl 2 > X2 3 act upon them as upon the elements of stress, 
and first examine this hypothesis when the three operators 
separately act on the six equalities in (2). 

Selecting the first equality in (2) and applying the 
operator Xlj on each of the two sides of the equality, 



then 



fljP = 



and ^6 2 ~d 2 0, 9o V*i 

d: 2 d/y 2 dyd- 

becomes 

2 y*i , o3 8 f. , «,y «?.-<?,) 2 B 2 (^-^s) 
9« 2 By 2 " ctyd^ d.yd^ 

B 2 



-.&-£)* 



which is null, and the hypothesis is not negated. 

In fact, the hypothesis is not negated in any case. 

The argument is then as follows : — If we had written 
did, n 2 #> etc. in the equations, and had proceeded to find 
these 18 quantities, we should have 18 linear equations 
from which to find them. The solution is therefore unique, 
and cannot differ from that employed by hypothesis and 
found to satisfy the 18 tests. 

We shall therefore regard 1? 6 2 i @z-> ^1? ^2? ^3 as acting 
as elements of a stress, though the} T have not the proper 
dimensions. 

It is proposed to find the conditions which must exist 
between these primary arbitrary stresses in order that the 
elements P, Q, R, S, T, U may be elements of an elastic 
stress. 



278 Mr. R. F. Gwyther on an Analytical Discrimination 

Note. — If, in the ordinary notation for strains, we give 
a, 6, c each one-half of" the usual value given to it, strains 
would follow the same laws of composition and resolution as 
stresses, and would therefore have the same differential 
operators. In this paper, I shall use a, b, c in this sense- 
that is, one-halt' of their usual. value. 



Lemma B. 

Except when we have reasons for keeping the expressions 
quite genera], it will suffice to limit the arbitrary stress- 
system to such stresses as have the co-ordinate axes as their 
principal axes. . " 

If in equation (2) the elements of stress, P, Q, etc., 
are made zero, the set of equations will then be recognized 
as indicating that 1 = e, 2 = f, 3 = g, ty 1 — a, ^—b, ^ 3 = c, 
where e, f, g, etc. are elements of strain arising from some 
arbitrary displacement. 

Hence on the right-hand side of equations (2) we may 
always replace 0\ by 0\ — e, 6 2 by 02— f, 0s hy B — g 9 "^i by 
"ty\ — a, yfr 2 by ty 2 --b, yfr s by t|r 3 — c. Consequently we may 
eliminate several sets of three functions, such as a/^, ty 2 , 
and i^ 3 , when some displacement is possible which makes, 
say, ^*i = a, ^ 2 = b, ^ 3 = c. 

Hence 






p-_^_^3 etc 



3y3* 



, etc., 



which is the form given by Airy, is a quite general form of 
solution, although for the purpose of this paper the full form 
given in (2) is requisite until we have decided upon some 
particular set of axes. 

2. The choice of a vector to represent the displacement, 
and the descriptive criterion of elastic stress. 

The mechanical stress has been represented in terms of an 
arbitrary stress-system, and it is possible and desirable to 
represent the displacement in terms of a similar stress- 
system. 

For this purpose I form a subsidiary stress-system, 



of Elastic Stresses in an Isotropic Body. 279 

indicated by 

fl.+fl 2 + fl 3 a Ol +Oi + O* n 0l + 0* + Oz n 

2 u 2 ' 2 3 ' 

This subsidiary system may be described as comple- 
mentary to the primary stress-system, in the sense that 
the two together form a hydrostatic pressure whose intensity 
is one-half of the sum of the principal stresses or one-half 
of the First Invariant of the primary stress-system. 

I shall form the assumed components of the displacement 
from the elements of this subsidiary stress in the manner of 
forming a force-system from a stress-system. 

Thus, I shall write 

2nw^-2^-2^ + ^ { 1 + 2 -0 s ). . . (3) 

Forming the values of S, T, U from these on the elastic 
stress-strain hypothesis we have 

s = a^i ay* aVs BVi_BVi 

'dy'dz ~dx~dy ~dx~dz ~dy 2 "dz 2 ' 
etc. 

On equating these to the values for the same elements given 
in (2), we find they require 

V s f = 0, V 2 t 2 =0, V s f 3 = 0. . . (4) 

Since {0 h 2 > $3» ^i> ^2? ^3} ac ^ on transformation 
of coordinates as elements of stress, it follows that the 
system must consist of a hydrostatic pressure and a general 
stress-system, each of the elements of which is a Spherical 
Harmonic. That is, 

WWe V 2 *i = 0, V 2 % 2 =0, V 2 %3 = 0. . . (5) 

This is the descriptive criterion of an elastic stress- 
system. 



280 Mr. R. F. Gwyther on an Analytical Discrimination 
3. Completion of the discrimination. The metric criterion. 

The remaining requirements of the stress-strain relations 
may be written 

\o% oyJ 

p +Q+B =(3»-„)g4; + ^). . . (6 ) 

which are to be completed from the values in (2) and (3). 
The first two only confirm the descriptive criteria of (4) 
and (5). The last leads to 

dj/0^ a^os d#dy J 






•im - n r 2 , „ » . f a^, d 2 2 ?i 2 



dyo^ d^o^ oocdy J J 
and therefore to 

(3m + 7i)V 2 (0i + 2 + 3 ) 

lor a«/ 2 oz 2 oylbz d#d- d^'dyJ 
or 

(m+n)S7 2 <j) 

( 0« 2 0«/ 2 O^ di/Oz 0*'02 0*02/ J 

• • • (7) 
which is the metric criterion and completes the dis- 
crimination sought for. 

This can be integrated, and gives <p in terms of the %'s 
and ^s_, of which we can always take the ty's to be null, 
when desirable. 

This completes the investigation for Cartesian coordinates 
under normal tractions only and with no inertia terms. 
With the conditions in (4), (5), and (6), the equations (2) 
give the general elastic stress- system, and these con- 
ditions discriminate an elastic stress-system from any other 
mechanical stress-system. 



of Elastic Stresses in an Isotropic Body. 281 

4. The inclusion of inertia terms. 

We must now modify equations (1) by writing pii, pv, 
pib on the right-hand side, where u, v, to are to have the 
values given in (3). 

We consequently replace P by P — p{9< 2 + 6 z — 6 x )\2n, 
Q by Q-K& + 01-00 /2», R by R- /0 (g 1 + g a _g 3 )/2n, 
S by S + /^h/», T by T+/njr 2 /w, U by U + p^/n. 

With these alterations the equations (2) still hold good. 

In forming our criteria, we equate values found from (2) 
to values given by the stress-strain relations deduced 
from (3). In these latter P, Q, R, S, T, U are to have 
the original values of these quantities and not those which 
replace them as above. 

Consequently, as our first step in the criteria, in place of 
V 2 ^i = etc. we obtain 

nV 2 ^i = piri, »Wa = H^, «V 2 fs = P^ 
and similarly 

™V 2 %1 = PXU n V 2 %2 = P%2, WV 3 'X8 = P%3- (8) 

In place of the last stage which gave the metric criterion, 
we find 

, o ^ 2 %i j 2 d 2%2 ; o ^ \ 



oyoz dx^z d^ctyj J 

and finally, 

+ ^^m-°- (9) 

These are the modified form of (5) and (7). 

5". Inclusion of bodily forces, with particular reference 
to gravity on the surface of the Earth, and to 
centrifugal forces. 

In any case we shall have to consider the alteration made 
in equations (1), and their solution in (2) by the introduction 



282 Mr. R. F. Gwyther on an Analytical Discrimination 

on the left-hand side of (1) of terms representing the com- 
ponents of the force per unit volume. 

These components can always be represented in a form 
similar to that given for the displacement in (3), but I shall 
suppose that the force per unit volume can be represented 
by the simpler forms 

B(F+/x) 5(F+/,) ^(F+/ 3 ) 

P ^5i~' p ^i~' p ~^~' 

and I shall suppose that we have selected the axes and 
that the yfrs are null. Then in (2) we must replace P 
by P + ^F+Z,), Q by Q + p(F+/,;, R by R + p(F+/ 8 ), 
leaving S, T, U unchanged. 

In the values found from the stress-strain relations there 
are no such changes to be made. 

We shall thus obtain from (6) 

V*0i-~tfi = V 2 2 - P f 2 = V 2 3 -pfi 
2n{3 /5 F4/>(/i+/>+y8)} + (3m + n)V!(^i + ^+.ft) 

= 6m {'S? + a? + a?}- ' (10) 

If we write 

^3 = </> + %3 + f + %/, 

where <£, % l9 ^ 2 , %3 have the values in (5) and (7) and may 
be regarded as Complementary Functions, then we remain 
with 

and 

2n/) F + (m + n>(/ 2 +/ 2 +/ s ) + (m + n) V 2 <£' 

which may be regarded as giving the Particular Integral 
corresponding to the particular force acting. 

There are not many cases of interest. In the case of 
gravity on the surface of the Earth, as under natural forces 
generally, we have 

j\ =j 2 =fi = and (m + n) V 2 </>' + 2npF = 0. 

If we suppose ( — X, — ^, —v) to be the direction-cosines 
of the attraction of gravitation, 

F ^^gfrt + fiy + pz) 

and ^ = _^ (x ^ + , + v ^. 



of Elastic Stresses in an Isotropic Body. 283 

and if we write P', Q', R' for the Particular Integral portion 
of P, Q, R — i. e. the terms which depend explicitly on g — 



we find 



11X — it 

?'=ffp\x + ~^gp(W + vz), . . . (12) 



with similar values for Q' and R', the Complementary 
Function part of P, Q, and R and the values of S, T, U 
being those given in the earlier part of this paper. 

The other case which I propose to consider is that of 
a body moving with angular velocities co x , co y , co z about 
the axes of coordinates which must be axes fixed in the 
body. It is implied either that the question is purely 
kinematica], or that a problem in Rigid Dynamics has 
been previously solved. 

The expressions for the acceleration of a point in the 
body are well known, and give for the effect of the reversed 
effective forces 

F = 

i{0)/ + »/> 2 + (w 2 + ft>*V+ (o> x 2 + a)/)* 2 

— 2(o x (D y xy — 2co x o)zxz — 2co lJ (t) y yz } , 
with 

f l = x(yw,—z6)y), fz—y{z<d x —!cwz), f 3 = z(xw y -y6) z ). 

The form of the forces / x , / 2 , / 3 indicates that they will 
cause no strain in the body, and consequently cause no 
stress. If we proceed to find the effect which they have on 
the values of the stresses, they will be seen to disappear from 
the stress-equations. I shall therefore omit them for this 
purpose, and treat %/, yj > X$ as nu ^- 

We then find 

*' = ~ 12(Jt n) { (< + W * 2) * 4 + (ft>/ + "^ 4 (W/ + "^ 

- 2co y co z yz(y 2 + z 2 ) - 2co x co z xz(x 2 + z 2 ) 

— 2(D x (0 y xy(x 2 + y 2 )} 

s '=Hi' ete -' w 

with Complementary Function terms as before. 

These give no solution of any specific question. They 
only give a skeleton of the general form which a solution 
will take. 



and 



[ 284 ] 

XXI. Note on Damped Vibrations. 
By H. S. RoWELL *. 

IT is well known that the space time curve for free un- 
damped vibrations may be derived from the projection 
of a rotating vector, the end of which describes a circle, 
and it is fairly well known that for vibrations which are 
resisted by fluid friction proportional to the velocity, the 
space time curve may be projected (as remarked by P. G. Tait) 
from a rotating vector, the end of which describes an equi- 
augular or logarithmic spiral. 

The vibration of bodies when resisted by a constant 
frictional force — say solid friction — is of great importance 
in practical work and does not appear to have been adequately 
treated. The results obtainable are, moreover, in themselves 
of much interest. 

If F is the constant force of friction the equation of 
motion is 

m'x+c 2 x±¥ — 0, 

wherein the sign of F depends on the direction of motion. 




Substituting a? = X+ F/c 2 , we have 



F 
X = x + -j = A cos 



\\/m ' 



which gives a series of harmonic vibrations about alternating 
centres distant F/c 2 from the equilibrium position when 
* Communicated by the Author. 



Notices respecting Neiv Boohs. 285 

friction is absent. The motion can be obtained by pro- 
jection from a spiral which is composed of semicircles as 
shown in the diagram. 

It will be seen that the amplitudes are in arithmetical 
progression and the difference for a complete period is 4F/c 2 , 
which may be called the arithmetic decrement. There is 
little purpose in using the ordinary definition of decrement, 
but it may be remarked that on this definition (i. e. ratio 
of successive amplitudes) the decrement ranges from unity 
for infinite amplitudes to infinity for zero amplitudes. 

The spiral curve described here does not appear to have 
been used before in scientific work, and it might be con- 
veniently called the arithmetic spiral or the spiral of semi- 
circles. 



XXII. Notices respecting New Boohs. 

Wmther Prediction by Numerical Process. By Lewis P. Richard- 
son, B.A., F.Inst.P. 4to, pp. xii + ^36. Cambridge Univer- 
sity Press. 30s. net. 

HPHE usual method employed in weather forecasting is a 
-*- development o£ that of Abercromby. Distributions of 
pressure are classified according to standard types, and the vari- 
ation on any occasion is predicted according to the behaviour of 
the atmosphere on previous occasions when conditions of the 
same type occurred. The method is therefore one of sampling 
inference in which the information utilized is all of one kind. 
Mr. Richardson believes that other information is relevant to the 
behaviour of the atmosphere ; and in this book he shows how to 
make use of the known results expressed in the hydrodynamical 
equations of motion and the equations of emission, transference, 
and absorption of heat and water. The method adopted is to 
work with equations each containing only one partial differential 
coefficient with regard to the time, so that this can be determined 
by means of the equation when the other quantities involved are 
known ; they include, of course, partial derivatives with regard to 
the position on the map and the height. These are to be found 
by observation at stations distributed according to a regular 
pattern, and the rate of change of each meteorological element at 
each station is to be calculated from them. Complications arise 
from the facts that the observations must be made at finite 
intervals both of time and of position, but these are allowed for. 
The stations required are more numerous than those at present 
in operation, and observations should be made every three hours 
to obtain the best results. Observations of upper-air winds and 
temperatures are required. 



286 Geological Society : — 

The method is one that appeals strongly to the mathematical 
physicist. It is necessarily laborious in its present form, and 
probably could not be worked with sufficient speed to make it a 
practical method of forecasting ; but when forecasters have 
acquired experience in its use, they will probably find that a 
sufficient number of the quantities allowed for are comparatively 
small to make it possible to expedite the calculation considerably 
without great sacrifice of accuracy. 

The value of the work is not confined to the application to 
forecasting, though the possibility of predicting the disturbing 
occasions when cyclones cause merriment in the daily press by 
moving in the wrong direction makes this the feature of most 
general interest. Its discussion of the physical properties of the 
atmosphere is so thorough that it constitutes a text-book of the 
subject. Copious references to original literature are given, and 
any meteorologist requiring serious information on any topic will 
do well to look first in this book. The section on evaporation 
suggests that the only limitation on the evaporation from vege- 
tation is imposed by the difficulty of passing along the stomata 
tubes ; this is not always true even for an isolated leaf, and is 
certainly wrong for a carpet of grass, on account of the obstruc- 
tion offered by the vapour from one stoma to evaporation from 
another. The numerical data actually given, however, eliminate 
this source of error. 

Concerning the printing and style of the book, it is only 
necessary to say that it is published by the Cambridge University 
Press. The index is good. H. J. 



XXIII. Proceedings of Learned Societies. 

GEOLOGICAL SOCIETY. 

[Continued from vol. xliii. p. 1138.] 

February 1st, 1922.— Mr. E. D. Oldham, F.K.S., President, 
in the Chair. 

Mr. Cyril Edward Nowill Bromehead, B.A., F.G.S., de- 
livered a lecture on the Influence of Geology on the 
History of London. 

The 6-inch Geological Survey maps constructed by the Lecturer 
were exhibited, and some of the new features pointed out. The 
small streams now * buried ' are indicated on the maps, and the 
historical research involved in tracing them led to an appreciation 
of the connexion between the geology and topography on the one 
hand, and the original settlement and gradual growth of London 
on the other. The reasons for the first selection of the site have 
been dealt with by several writers : below London the wide allu- 
vial marshes formed an impassable obstacle ; traffic from the 



Influence of Geology on the History of London. 287 

Continent came by the ports of Kent, and, if destined for the 
north or east of Britain, sought the lowest possible crossing of 
the Thames. This was near old London Bridge, where the low- 
level gravel on the south and the Middle Terrace deposits on the 
north approached close to the river-bank. A settlement was 
obviously required here, and the northern side was chosen as the 
higher ground. The gravels provided a dry healthy soil and an 
easily accessible water-supply ; they crowned twin hills separated 
by the deep valley of the Walbrook, bounded on the east by the 
low ground near the Tower and the Lea with its marshes, and on 
the west b}' the steep descent to the Fleet ; the site was, therefore, 
easily defensible. The river-face of the hills was naturally more 
abrupt than it is now, owing to the reclamation of ground from 
the river ; the most ancient embankment lay 60 feet north of the 
northern side of Thames Street. 

The first definite evidence of a permanent settlement was the 
reference in Tacitus. The early Roman encampment lay east of 
the Walbrook, and the brickearth on the west around St. Paul's 
was worked. Later the city expanded, until the St. Paul's hill 
was included, the wall being built in the second half of the 4th 
centuiy. The great Roman road from Kent (Watling Street) 
avoided London, and utilized the next ford upstream — at West- 
minster — on its way to Yeralamium and the north-west, The 
earliest Westminster was a Roman settlement beside the ford, 
built on a small island of gravel and sand between two mouths of 
the Tyburn. This settlement could not grow, as did London, 
since the area of the island, known to the Saxons as Thorney, was 
small. The road from London to the west joined the St. Alban's 
road at Hyde Park Corner, running along the ' Strand,' Avhere the 
gravel came close to the river ; a spring thrown out from this 
gravel by the London Clay was utilized for the Roman Bath in 
Strand Lane. 

Throughout Mediaeval times London was practically confined to 
the walled city, a defensible position being essential. The forests 
of the London-Clay belt on the north are indicated in Domesday 
Book and referred to by several writers, notably Fitzstephen, 
whose Chronicle also mentions many of the springs and wells 
and the marsh of Moorfields, produced largely by the damming 
of the Walbrook by the Wall. The same writer mentions that 
London and Westminster are ' connected by a suburb.' This 
was along the ' Strand,' and consisted first of great noblemen's 
houses facing the river and a row of cottages along the north 
side of the road ; this link grew northwards, at first slowly, 
but in the second half of the 17th century with great rapidity. 
By the end of that period the Avhole of the area covered by the 
Middle-Terrace Gravel was built over, but the northern margin of 
the gravel was also that of the town for 100 years, the London- 
Clay belt remaining unoccupied. 

The reason for this arrested development was that the gravel 



288 Intelligence and Miscellaneous Articles. 

provided the water-supply. In early days the City was dependent 
on many wells sunk through the gravel, some of which were famous, 
such as Clerkenwell, Holywell, and St. Clement's. In the same 
way the outlying hamlets (for instance, Putney, Roehampton, 
Clapham, Brixton, Ealing, Acton, Paddington, Kensington, 
Islington, etc.) started on the gravel, but later outgrew it, as 
pointed out by Prestwich in his Presidential Address of 1873. In 
the City the supply soon became inadequate, or as Stow says 
' decayed,' and sundry means were adopted to supplement it. The 
conduit system, bringing water in pipes from distant springs, began 
in 1236 ; London-Bridge Waterworks pumped water from the 
Thames by water-wheels from 1582 to 1817 ; the New River was 
constructed in 1613, and is still in use. It was not until the 
19th century that steam-pumps and iron pipes made it possible 
for the clay area to be occupied, thus linking together the various 
hamlets that are now the Metropolitan Boroughs. 

Some of the ways in which Geology affects London to-day were 
briefly indicated, and the lecture was illustrated by a number of 
lantern-slides, reproduced mainly from old maps and prints. 



XXIV. Intelligence and Miscellaneous Articles. 
young's modulus and poisson's eatio for spruce. 

To the Editors of the Philosophical Magazine. 
Dear Sirs, — 
IN my recent paper in the Philosophical Magazine for May 1922 
-*- there is an error on page 877. It is there stated that 

G"yz Gzy G"zx °~xz °~xy Gyx 

TP ' TH TH 5 XT* T? 

y ih z Jh z & x \h x Jbjy 

S 
are equal respectively to 700 S«, 700 S^ and ~ . 

£>x 

This should read 

E z _ Ojy E^ _ OjX Ea; _ Gjy 

Ey (TyS E z <T X z Ey (T yX 



S 
are equal respectively to 700 S«, 700 8 X and ~. 

The error becomes evident on reading the paper, but I xery 
much regret that it has crept in. 

Tours faithfully, 
The College of Technology, " H# CABBiwcwxMr. 

Manchester. 

May 24th, 1922. 



Tizahd & Pie. 




Corresponding to experiment A 



Fig. 2. 




Corresponding to experiment A 



In these photographs the lower horizontal line is the line 
of atmospheric pressure. The ordinates represent pressure, 
and the abscissae time. A is the beginning of compression, 
B the point of maximum compression, and C the explosion. 
The curve in the top left-hand corner is the cooling curve 
of the products of combustion. 



Phil. Mag. Ser. 6, Vol. 44. PI. 



FlG. 3. 




Corresponding- to experiment A 
Fig. 4. 




Corresponding to experiment Ds 
Fig. 5. 




Corresponding to experiment D-. 



THE 
LONDON, EDINBURGH, and DUBLIN: 

PHILOSOPHICAL MAGAZINE 

AND 

JOURNAL OF SO 

[SIXTH SEIU 



A UGUST 1922. 




XXV. On the Viscosity and Molecular Dimensions of Gaseous 
Carbon Oxysulphide (COS). By C. J. Smith. B.Sc, 
A.R.C.S., JD.I.C, Research Student, Imperial College of 
Science and Technology*. 

'TPHE present research is a continuation of the work on 
JL the measurements of the viscosities of gases, for the 
purpose of elucidating the structure of the molecules 
constituting them. Some measure of success has attended 
this investigation in many cases where the necessary data 
are known, and suggests that an accumulation of further 
similar data may be fruitful. A case in point is that of the 
molecule of carbon oxysulphide, and this paper describes the 
measurements of the viscous properties of this substance. 
which is ordinarily gaseous. The data, hitherto unknown, 
which have been obtained, have been applied to calculate the 
molecular dimensions in the ordinary way. 

Apparatus and Method of Observation. 

The apparatus and method, which have been used to 
determine the viscosity of carbon oxysulphide, have recently 
been fully described f. 

Method of Experiment. 

The viscometer was carefully standardized with a new 
mercury pellet in the manner indicated in previous papers. 

* Communicated bv Prof. A. 0. Ranlrine, D.Sc. 

t A. 0. Rankine and C. J. Smith, Phil. Mao-. vo l. xlii. p. 60], Nov. 
1921, and C. J. Smith, Proc. Phys. Soc. vol. xxxiv. p. 155, .June 1922. 

Phil. Mag. S. 6. Vol. 44." No. 260. Aug. 1922. U 



290 Mr. C. J. Smith on the Viscosity and Molecular 

and the corrected time o£ fall proved to be 104*70 sees., a 
value which is probably correct to 0*1 sec. With this time 
of: fall the corresponding time of fall for carbon oxysulphide 
has been compared, and with appropriate corrections gives 
the relative viscosity, from which the absolute viscosity has 
been obtained by assuming the viscosity of air at 15 o, C. 
to be 1*799 x 10" 4 C.Gr.S. units. In addition, the variation 
of viscosity with temperature has been derived from com- 
parisons of the corrected times of fall at atmospheric and 
steam temperature. 

Preparation and Purification of the Carbon Oxysulphide. 

The carbon oxysulphide was prepared by the action of 
sulphuric acid (five vols, acid, four vols, water) on pure 
potassium thiocyanate in the cold (room temperature). At 
the same time hydrocyanic acid, formic acid, and carbon 
bisulphide are formed. To remove these impurities the 
method recommended by Moissan * was used. This consists 
In passing the gas through a strong solution of caustic 
potash to remove the hydrogen cyanide and then over wood 
charcoal to remove the carbon bisulphide. The gas was 
dried by being passed over calcium chloride, and then 
solidified at liquid air temperature. All permanent gases 
were pumped out of the U-tube containing the solid COS by 
means of a mercury pump. The liquid air was then replaced 
by a mixture of solid 00 2 and alcohol at —80° C, when it 
was observed that the vapour pressure of the liquid COS was 
about 30 cm. of mercury. The C0 2 mixture was then 
removed and samples of the gas collected over mercury. It 
was further purified before being introduced into the 
viscometer by fractional distillation at liquid air temperature. 
The liquid air having been replaced by C0 2 and alcohol at 
— 80° C, it was possible on account of the comparatively 
high vapour pressure of COS at this temperature to pump 
off successive quantities of dry COS sufficient io fill the 
viscometer at atmospheric pressure. 

Experimental Results. (Table I.) 

W^e have -f 15 = 69*96 sees., and £ 100 — 90*64 sees. 
The ratio of these times of fall gives the ratio of the 
viscosities at the corresponding temperatures. 
Thus 

*7ioo _ ^ioo _ 9 0*64 _-,. 9 QKp 
Vis ~ *i5 69 96 ~ 
* Moissan, Traite de Chimie, vol. ii. p. 318. 



Dimensions of Gaseous Carbon Oxysulphide. 291 

Table I. 
Each time recorded is the mean of four observations in each 
direction for the whole pellet, and of three for the pellet wh 
divided into two segments. The letters in parent heses indicate t 



order in which the observations were made. 



en 
the 



Temp. 

(deg. C). 


Time of fall (sees.). 


Capillary 

correction 

(.r). 


Corrected 

time 

(0- 


Time at 


Whole 
pellet. 


Two 

segments. 


15°0C. 


ioo°-oc. 


(a) 1529... 


73-04 


7635 


00416) 


7001 


69-92 




(6) 15-44... 


73-08 


76-25 


00399 


70-16 


7005 


— 


(e) 15-63 ... 


7302 


7622 


0-0403 


70-07 
Mean 


69-91 
69-96 


— 


(c)99-9 ... 


91-51 


92-43 


0-0099 


90-60 


— 


9063 


(rf)99-9 ... 


91-54 


92-48 


o-oioi 


90 62 
Mean 


— 


90-65 
9064 



Assuming Sutherland's law to hold for this gas, the value 
of Sutherland's constant obtained is C = 330. 
Also at 15°'0 C, 

/cos = 69'96 
4 ir 104*71) 



= 0-6682. 



Correcting for difference of slipping of COS and ah\, we 
obtain 

259i= 0-6668. 

fail- 
Oil the assumption that the viscosity of air at 15°0 C. is 
1-799 x 10- 4 C.G.S. units, the values for COS are 

Vl .= 1-200 x 10" 4 C.G.S. units 
and ?7 100 = 1*554 x 10" 4 C.G.S. units ; 

and by extrapolation using Sutherland's formula, 
^=1-135x10- 'C.G.S. units. 

Calculation of Molecular Dimensions. 

The particular dimension calculated from the above 
results is the mean area A which the molecule presents in 
mutual collision with others. The basis of this calculation 
is Chapman's formula (Joe. cit.) modified in its interpretation 
in the manner suggested by Rankine. The value obtained 
is A = 106 x 10 _ir ' cm. 2 which may be subject to an error of 
2 or 3 per cent. 

U2 



292 -Prof. A. 0. Rankine on the Molecular Structure 
Summary of Results. 
Table II. 



Viscosity in C.G.S. units X 10-4. 

Sutherland's 


Mean 
collision area. 
em. a x'10— 1B . 


15°0C. 


100°'0 C. 


constant. 

o°-oc. 

i 


1-200 


1-554 


1-135 330 


1-06 



In conclusion the author gladly acknowledges the grant 
for this research, which was made by the Government Grant 
Committee of the Royal Society, and also wishes to thank 
Professor Rankine for his continued help and advice. 

Imperial College of 

Science and Technology, S.W. 7. 

1st May, 1922. 



XXVI. On the Molecular Structure of Carbon Oxy sulphide 
and Carbon Bisulphide. By A. 0. Rankine, D.Sc, 
Professor of Physics in the Imperial College of Science and 
Technology*. 



1. FTHHERE are at the present day in the process of 
JL development several theories of atomic and mole- 
cular structure which are in many respects discordant. 
They have, however, at least one feature of general agree- 
ment — namely, the common view that the atoms of the 
inert gases occupy unique positions in the various schemes. 
The distribution of the electrons with respect to the nuclei 
in these atoms is regarded as having the characteristic of 
completeness, so that there is displayed no marked tendency 
to lose electrons or to capture additional ones. Moreover, 
atoms other than those mentioned are believed to have in 
varying degrees what may be called deficiencies and 
redundancies of extra-nuclear electrons, which they endeavour 
to adjust by entering into suitable combinations with one 
another ; so that either by the process of give and take, or 
by common use of the same electrons, con figurations corre- 
sponding closely to those of the inert atoms are attained bv 
the individual atoms forming the compound. 

2. These views of chemical combination find their most 

* Communicated by the Author. 






of Carbon Oxysulphide and Carbon Bisulphide. 293 

complete expression in the theory of Lewis unci Langmuir *, 
particularly in relation to the type of compound with which 
this paper is concerned — namely, that in which atoms, 
deficient in electrons, are regarded as sharing- them in order 
to reach the completeness of inert configurations. The main 
purpose of this paper is to apply the principles of this theory 
to the special case of the molecule of carbon oxysulphide, 
and to show that the molecular dimensions of this compound, 
as derived from viscosity data, are consistent with the Lewis- 
Langmuir view of its constitution. This test of the validity 
of the theory is made possible by the recent measurements 
by 0. J. Smith t of the viscous properties of the gas in 
question. Similar calculations for the molecule of carbon 
bisulphide have been made, and these await verification or 
otherwise when the necessary viscosity data are available. 

3. Carbon oxysulphide belongs to a family of three 
compounds having the chemical constitutions C0 2 , COS, and 
CS 2 . The two former are gaseous at ordinary temperatures, 
and the latter a highly volatile liquid. In all of them carbon 
is a constituent, and COS can he regarded as the molecule 
obtained by the substitution of a sulphur atom for one of the 
oxygen atoms in C0 2 , or by the reverse substitution in CS 2 . 
It is probable that the carbon atom occupies the central 
position in each molecule, and that the nuclei of the three 
atoms lie in each case upon a straight line. 

4. According to the Lewis-Langmuir theory (loc. cit.), the 
atoms in these molecules are linked together by sharing 
external electrons in such a manner that each atom approxi- 
mates to the configuration of the inert atom at the end of the 
corresponding row in the periodic table. Thus, in C0 2 the 
central carbon atom shares altogether eight electrons, four on 
each side with an oxygen atom. The electron configuration 
thus formed is that of three neon atoms in a row, for the 
inert atom corresponding to both carbon and oxygen is neon. 
In the molecule COS there are again eight electrons shared 
by the carbon atom, four on one side with the oxygen atom, 
and four on the other side with the sulphur atom. The 
electron arrangement thus attained is that of two neon atoms 
(corresponding to the oxygen and carbon) and one argon 
atom (corresponding to the sulphur). Applying a similar 
argument to the CS 2 molecule, we are led to regard it as 
resembling closely the electron distribution of inert atoms 
in the sequence argon-neon-argon in a line. In other 
words, we can treat each carbon or oxygen atom in a 

* I. Langmuir, Joura. Amer. Chem. Soc. vol. xli. p. 8C>8. 
t C. J. Smith, supra, p. 289. 



294 Prof. A. 0. Rankine on the Molecular Structure 

molecule as having nearly the same dimensions as a neon 
atom, and each sulphur atom in combination as approxi- 
mating to the dimensions of an atom of argon. 

5. The remaining question of how far apart are the nuclei 
of the atoms in the molecule finds a satisfactory answer in 
the work of W. L. Bragg*, whose X-ray -crystal measure- 
ments have enabled him to assign probable values for the 
radii of the outer electron shells of the atoms of the inert 
gases. The only values with which we are at the moment 
concerned are those of neon o and argon, which are given 
respectively as 0*65 and 1*03 Angstrom units. In cases like 
those under consideration, where outer electrons are playing 
a double part, the sharing is equivalent to contiguity of the 
outer shells, so that the distance apart of the nuclei is the 
sum of the radii of the appropriate inert atom shells. Thus 
for C0 2 , which is pictured as three neon atoms in line, the 
three nuclei are equally spaced and separated by distances 
equal to twice the radius of the neon outer shell, i. e. 
2x0-65 A = 1'30 A. In COS the distance between the 
carbon and oxygen nuclei is the same, namely 1*30 A, 
but the distance between the carbon and sulphur nuclei 
is the sum of the radii of the outer electron shells of 

O Q O 

the neon and argon atoms, i. e. 0*65 A-f 1*03 A = 1*68 A. 
The three nuclei in COS are thus unequally spaced on 
account of the greater size of the argon atom. In CS 2 the 
distance between the o carbon nucleus and each sulphur 
nucleus is also 1*68 A, and the three nuclei are again 
spaced symmetrically. 

6. It is evident that none of the three molecules under 
consideration, if their configurations are as indicated, can be 
expected to display spherical symmetry. In these circum- 
stances it is necessary to interpret in a special way the 
results of the well recognized method of calculating 
molecular dimensions from viscosity data. The quantity 
which is actually derivable from the formula is the mean 
value of the area which the molecule presents, for all 
possible orientations, as a target for mutual collision with 
other molecules in the gas. This area the author f has 
ventured to call the mean collision area, and its value for 
COS is given by C. J. Smith (loc. cit.) as 1*06 x lO" 15 cm. 2 
The immediate problem before us is to find how nearly the 
tentative model of this molecule described above would 
exhibit this value for its mean collision area. The values of 

* W. L. Brag-o-, Phil. Mag. vol. xl. p. 169. 

| A. O. Kankine, Proc. Eov. Soc. A, vol. xcviii. p. 360, and Proc. 
Phys. Soc. vol. xxxiii. p. 362. 



of Carbon Oxy sulphide and Carbon Bisidplride. 295 

the mean collision areas of the constituent configurations 
(which we are taking- to be those of neon and argon) are 
known, and it is usual to regard these symmetrical inert 
atoms as behaving as clastic spheres for purposes of collision. 
The radii of these collision spheres, as we may call them 
for the sake of precision, are 1*15 A and 1*44 A respectively, 
and they are considerably larger than those of the corre- 
sponding outer electron shells, so that they overlap when 

Fig. 1. — Molecular Dimensions from the point of view 
of the Kinetic Theory. 




The Carbon Dioxide Molecule: equivalent to three linked atoms of Neon, 




The Carbon Oxysulphide Molecule : equivalent to two Neon atoiris and 
one Argon atom linked together. 




The Carbon Bisulphide Molecule : equivalent to two Argon atom® 
linked together by one Neon atom. 

the nuclei are separated by the distances demanded by 
electron sharing. Fig. 1 shows three models, drawn to scale, 
representing what we may conceive C0 2 , COS, and 0S 2 to 
be like for purposes of intermolecular encounters. C0 2 may 
be regarded as three overlapping spheres,- each of the neon 



296 Prof. A. 0. Rankine on the Molecular Structure 

collision size, with centres separated by the distances already 
•specified. In COS we take instead of one of the extreme 
neon spheres an argon collision sphere ; while in CS 2 both 
the outer spheres are of the argon size. In all three cases 
the diagram represents all the nuclei in th.Q plane o£ the 
paper, and the line joining them is evidently an axis of 
symmetry. 1£ these symmetrical axes are variously oriented, 
the area presented by the model assumes different values, 
and our problem is to calculate the mean value o£ this pro- 
jected area for comparison with that deduced from viscosity 
data. The author (loc. cit.) has already derived the necessary 
formulae for this purpose, and has shown that the result 
obtained by application to the first model in fig. 1, namely 
00 2 , is very nearly equal to the actual mean collision area of 
the carbon dioxide molecule. In other words,, a carbon 
dioxide molecule behaves in collision as though it had the 
configuration of three neon atoms in a straight line and with 
outer electron shells contiguous. 

7. Calculation for the COS Model. — In the model which 
we are taking to represent the COS molecule, the calculation 
in the strictest sense is greatly complicated by reason o£ the 
particular distribution of the spheres. The exact formulae 
which have been obtained {loc. cit.) for equal and unequal 
spheres only apply rigidly to cases where a special relation 
exists between the diameters of the spheres and the distances 
apart of their centres ; and the model under consideration 
does not fulfil this condition. But by regarding the problem 
from two different points of view, we can obtain, by means 
of the comparatively simple formulae already available, upper 
and lower limits which are so close together as to render 
unnecessary the laborious exact calculation. This course is 
all the more justifiable because it is fully recognized that 
the general treatment of the problem itself can only be taken 
as a first approximation to the truth. 

8. Let us consider the effect on the area of projection of 
the model (reproduced in the full lines of fig. 2, a) caused 
by variations of orientation of the symmetrical axis joining 
the centres ]5 2 , and 3 of the constituent spheres. It 
will be convenient to speak of the sphere with centre L 
simply as sphere 1, and so on, and of the projections of the 
spheres, which will of course be circles, as projection 1 
etc. As the axis 0, 3 approaches the line of sight, the 
projections of the centres approach one another, and the 
eclipsing of the spheres becomes more and more marked. Up 
to a certain point the total projected area is equal to the sum 



of Carbon Oay sulphide and Carbon Bisulphide, 297 

o£ the areas of the whole of projection 3, the crescent formed 
by the overlapping of projection 3 over projection 2, and the 
crescent formed similarly by the eclipse of projection 1 by 
projection 2. Before the eclipsing of 2 by 3 is complete, 
however, projection 3 begins to encroach upon regions of 
projection 1 which are not already covered by projection 2. 
Tt is this fact that introduces into the exact treatment of the 
problem the complications to which reference has already 
been made. Thus in fig. 2, b, which shows the projected 
area for that orientation of the axis for which the eclipse of 
2 by 3 is just complete, the crescent formed by 2 and 1 still 

Fiff. 2. 




•survives, but parts of it (as indicated by the shading) are 
covered by projection 3. The projected centres are 0/, 2 ', 
and (V respectively, and this particular state of affairs occurs 
when the angle between C^ 3 and the direction of projection 
is 9° 47' for the spheres having the dimensions and distribu- 
tion already specified. 

9. Overlapping of the type just indicated, like all 
overlapping, has the effect of reducing the projected area ; 
it is therefore clear that if w r e neglect it we shall obtain too 
large a value for the mean area of projection — that is, an 
upper limit will be obtained by taking the mean collision 
area as the sum of the three parts : (a) the area of the circle 
3, (It) the mean value of the area of the crescent formed by 
^cireles 3 and 2 (c) the mean value of the area of the crescent 



298 Prof. A. 0. Rankine on the Molecular Structure 

formed by circles 2 and 1. The first of these quantities is- 
the area of the central cross-section of the argon sphere 
itself, viz. 0*648 x 10 -15 cm. 2 ; the two latter are readily 
obtained from the graph in the paper already mentioned *.. 
They prove to be 0*217 x 10~ 15 cm. 2 and 0*226 x 10~ 15 cm. 2 
respectively. The total is 1*09 x 10 ~ 15 cm. 2 , and this provides 
our upper limit. 

10. With regard to the lower limit, we can obtain a 
satisfactory value by contemplating a variation of our model, 
which avoids the special type of overlapping responsible for 
complications. A suitable change for this purpose is to 
substitute for the sphere 1 a smaller sphere having the same 
centre but of such magnitude that its projection becomes 
just eclipsed by projection 2 at the same orientation of the 
symmetrical axis for which projection 2 is just eclipsed by 
projection 1, as shown by the dotted circles in fig. 2. o The 
radius of the necessary sphere is found to be 0*93 A as 
compared with the original value 1*15 A. Examination of 
the projection of a sphere of this size, in relation to the other 
two projections, shows that for no orientation does eclipsing 
of the shaded type appear, and the formulae already available 
enable the mean area of projection to be calculated exactly.. 
The value so obtained will, however, obviously be less than 
the true value aimed at, on account of the reduction of size 
assumed for sphere 1. Using the graph already mentioned,, 
the lower limit thus derived is 

0*648 x 10" 15 cm. 2 -f 0*217 x 10~ 15 cm. 2 + 0*138 x 10" 15 cm. 2 

= l-00xl0- 15 cin.* 

11. The foregoing justifies the assertion that a molecular 
model having the dimensions of an argon atom succeeded 
by two neon atoms in line and spaced according to the- 
demands of outer electron contiguity may be expected to* 
have a mean collision area intermediate between 

1*09 x 10- 15 cm. 2 
and 1-00 xlO- 15 cm. 2 

The actual value of the mean collision area of the COS- 
molecule, as determined from viscosity is 



with a possible error of 2 or 3 per cent. It falls definitely 
between the upper and lower limits obtained from our 
calculations, and seems to provide striking corroboration of 

* A. O. Rankine, Proc. Phys. Soc. vol. xxxiii. p. 371. 



of Carbon Oxysulphide and Carbon Bisulphide, 299 

the theory upon which the estimates are based. But we 
must be content with the conservative remark that the 
dimensions of the carbon oxysulphide molecule, as found by 
the application of the kinetic theory, are consistent within 
the limits of experimental accuracy with the view that the 
three atoms of the molecule, by sharing external electrons, 
assume the electron configurations and behaviour in collision 
of particular groupings of the neighbouring inert atoms. 

12. Calculation for the CS.> Model. — Although there exist 
at present no data for carbon bisulphide which suffice to 
calculate the mean collision area of the molecule in the 
gaseous state, the success of the previous comparison 
would appear to justify a prediction of its value by con- 
sideration of the appropriate model. This has been repro- 
duced in the full lines of fig. 3, a. Here again the model is 




one which does not lend itself to exact solution without 
laborious calculation ; but again, also, we can obtain 
satisfactorily close upper and lower limits. The area 
of projection will clearly be less than that corresponding to 
the model in which the dotted sphere is substituted for the 
small central one, so that we have three equal spheres of 
the argon size in line : it will, on the other hand, be greater 
than if the central sphere is entirely dispensed with, so that 
there are two equal argon spheres only, as represented in 
fig. 3, b. The dimensions of the spheres and the distances 



300 Mr. F. P. Slater on the Rise of 

apart of their centres have already been stated ; and both 
modified models have mean areas of projection which are 
very easily calculated. The upper limit thus determined 
proves to be 2*12 times the collision area of the argon atom ; 
the lower limit is 1*90 times the same area. Using the 
known value 0*648 xlO -15 cm. 2 for the collision area of the 
argon atom, we find that the mean area of projection of the 
model consisting of two argon atoms with an intermediate 
neon lies between 

1*37 x 10" 15 cm. 2 

and l'23xl0- 15 cm. 2 

We may venture to predict with some confidence that the 
mean collision area of the CS 2 molecule, when determined, 
will be found to be between the above values. A more 
exact estimate could of course be made, but the degree of 
accuracy at present attainable in determining molecular 
dimensions from viscosity measurements is not sufficient to 
render the additional calculation worth while. 

Summary. 

On the assumption of the validity of the Lewis-Langmuir 
view of molecular constitution, the probable behaviour during 
encounters has been examined for the molecules of carbon 
oxysulphide and carbon bisulphide. In the former case it is 
shown that the molecular dimensions as derived from the 
application of the kinetic theory to the viscosity measure- 
ments of C. J. Smith, are in striking accordance with the 
results of the above examination. In the latter case 
comparison is not yet possible, on account of the absence of 
necessary data. 

Imperial College of Science and Technology, 
May 11th, 1922. 



XXVII. The Rise of <y-Raij Activity of Radium Emanation. 
By F. P. Slater, MISc. (Vict.), B. A. [Cantab.)*. 

XN a previous paper t it has been shown how the initial 
rise of 7-ray activity, starting from pure radium 
emanation, depended on the nature of the walls of the tube 
containing the gas, the reason . being that a small but 

* Communicated by Prof. Sir E. Rutherford, F.R.S. 
t Slater, Phil. Mag. vol. xlii. p. 904 (1921). 



y-Ray Activity of Radium Emanation. 301 

detectable y« radiation was excited in the walls by the impact 
of the a. particles emitted by the emanation. The amount of 
this excited radiation was, however, very small when the 
walls of the tube were composed of atoms of low atomic 
weight, and for a lining of pure paper the 7-ray activity of 
the emanation and its products was found to rise practically 
from zero. Under such conditions the 7 radiations from 
the tube are due only to the products radium B and 
radium C. 

Taking the number of emanation atoms disintegrating per 
second at initial time as unity, the number of radium-B atoms 
disintegrating per second at any subsequent time t is 



X 2 \ s 2 



-Kit 



\ = 1,2,3 

where \ 1? X 2 , A, 3 are the transformation constants of the 
emanation and the products A, B, and C respectively. This 
quantity is tabulated for various times up to 220 minutes at 
the end of this paper (Table II.). 

Similarly, the number of radium-C atoms disintegrating 
per second at time t is 

\= 1, 2.3,4 

Tables for this quantity for various times up to 258 
minutes have been given by Moselev and Makower * and by 
Rutherford t- 

The rise in 7-ray activity of a tube filled initially with 
pure emanation can therefore be represented by 



KXoAs^T 



e-M 



(A 2 -X 1 )(X 3 -X 1 ) 
\ =1,2,3 



+ (1-K)A 2 \ 3 \ 4 g e ~ Xl * 



1,2, 3, 4 

where K is the fraction of the ionization, measured under 
given absorption conditions, due to radium B when in radio- 
active equilibrium with radium C. 

Thus it is necessary to determine "K." Since the 7 rays 

•• Moselev and Makower, Phil. Mag. vol. xxiii. p. 302'(1912). 
t Rutherford, 'Radioactive Substances/ p. 490. 



302 



Mr. F. P. Slater on the Rise of 



from radium B are less penetrating than those from radium 
C, " K " depends on the thickness of matter through which 
the radiations pass before entering the ionization chamber. 
Rise curves have been experimentally determined for 
different thicknesses of absorption material, both lead and 
aluminium being used. The values of K for various thick- 
nesses have been deduced by trial, and are shown in fig. 1. 



Fig. 1, 



0-30 

V 



f - 



;\ i i I 



> Mms. cf /ead. 

A comparison of the experimental and calculated rise curves 
of the 7~ray activity through 12'0 mm. of lead is given 
in fig. 2, After six minutes from the introduction of pure 
emanation, the calculated and experimental curves agree 
very closely. 

From these curves the absorption coefficient of the 
radium B-7 rays can be deduced, and the values found are 
given in Table I. along with comparative determinations 
by Makower and Moseley (loc. cit.) and Rutherford and 
Richardson *. 

The values of the absorption coefficients for the thick- 
nesses of aluminium are somewhat doubtful, since the 
supposition of homogeneity of the radium-C 7 rays is not 
justifiable through such small thicknesses. The increasing 

* Rutherford and Richardson, Phil. Mag. vol. xxv. p. 722 (1913). 



y-.Ray Activity of Radium Emanation. 



303 



value of fi (cm. -1 ) with decreasing thickness of absorption 
material (see Table I.) is to be expected, since Rutherford 
and Richardson (loc. cit.) showed that radium B emits 
certainly two types of radiation having absorption coefficients 
in aluminium of 0*51 cm, -1 and 40*0 cm. -1 , and possibly a 
third type (/a=230 , 0~ 1 in aluminium). 

Fig. 2; — liise of y activity from Radium emanation through 
]2'6 mm. of lead. 



%Max. 




I 






f 


Activity 

T 












1 


C-30 












I, 






Ex-perimem 


-a/ ri e 




/ 












I; 


0-50 












/! 










fi 




0-40 










i 












f; 




0-30 










/ 










/ 






0-20 








/,' 


























/> 












V 






0-10 






/,' 










^/s' 


■■'' 












^' 






Y 77/77<P /o 


m/hs. 



The absorption coefficients in lead only, given in Table I., 
are corrected for obliquity of the rays entering the electro- 
scope, and King's correction is used as given in Case II. 
of his paper *, 

I* /O0- cos 0/O* sec 0] 



I 



1 — cos 



where 1^ and I are the intensities of the radiation emerging 
through a plate of thickness t cm. and incident radiation 
respectively, /u, the absorption coefficient expressed in cm." 1 , 
and 6 the semi-angle of the cone of rays entering the 
electroscope. 

vol. xxiii. p. 248(1912). 



304 Rise ofy-Ray Activity of Radium Emanation. 

Table 1. , 
Absorbing medium is Lead, except where otherwise shown. 



Thickness of 
Absorbing Plate. 



Value of jw (cm.— i) Moseley Rutherford 

for and and 

Radium-B rays. Makower. Richardson. 



160-200 mm 


2-7 cm.-l 


_ 


Varying from 


100-150 ,, 


2-8 cm -i 


.... 


11-0 cm.-l 


4-0- 6-0 „ 


4-1 cm.- 1 


4-0 cm.-l 


to 






(lead). 


2-8 cm.-l in 


1-5- 2-0 „ 


6-2 cm.-l 


6-0 cm.-l 


lead. 


3-0- 4-0 „ (Aluminium). 
- 75 mm. (Aluminium) ... 


1-7 cm.-l (Al) 

10-0 cm.-l (Al) 


(lead). 
? 





Table II. 

Rise of Radium B from Radium Emanation. 

Maximum = 0*97480 is taken as unity. 



Time 
in 
mins. 
1 


Calculated 
rise of 

Radium B. 
.. 0-00269 
.. 0-01016 
.. 002129 
.. 003513 
.. 005115 
.. 0-06882 
.. 0-08718 
.. 010647 
.. 0-12622 
.. 0-14624 
.. 0-16637 
.. 0-18641 


Time 

in 
mins. 

14.. .. 

16 

18 

20 

30 


Calculated 

rise of 
Radium B. 
... 0-2260 
... 0-2649 
... 03023 
... 0-3379 
. . . 0-4942 


Time 
in 

mins. 
110 


Calculated 

rise of 
Radium B. 
... 0-9502 


2 


120 

1W 


.. 09643 


3 


... 0-9750 


4 


140 

150 

160 

170 

180 

190 

200 

210 

214 

220 


... 0-9835 
... 0-9886 


6 

7 

8 


40 

50 

<0 

70 

80 

90 ..... 
100 


... 0-6135 
... 0-7066 

... 0-7780 
.. 0-8329 
... 0-8748 
... 0-9058 
... 0-9315 


... 0-9929 
... 0-9956 
... 0-9979 


9 

10 

11 

12 


... 0-9992 
... 0-9998 
... 0-9999 
... 1-0000 
... 0-9999 



Summary. 

Curves showing the rise of 7-ray activity from pure 
radium emanation measured through a wide range of 
absorption thickness of matter have been determined and 
utilized in deducing the absorption coefficients of the hetero- 
geneous 7 radiation from radium B. 

My thanks are due to Professor Sir E. Rutherford for his 
invaluable help in carrying out this research, and to 
Mr. G. A. R. Crowe for the preparation of the radioactive 
material. 



[ 305 ] 

XXVI IT. An Experimental Test of Sm oluc ho iv ski's Theory of 
the Kinetics of the Process of Coagulation. By Jnanendra 
Nath Mukherjee, D.Sc, Professor of Physical Chemistry, 
University of Calcutta, and B. Constantine Papacon- 
STANTINOU, D.Sc, Assistant Professor of Chemistry, Uni- 
versity of Athens *. 

A short account of the Theory. 

IN some experiments on the degree of dispersion of 
colloidal arsenious sulphide on the rate of coagulation, 
it has been shown (J. Amer. (Jhem. Soc. vol. xxxvii. p. 2026, 
1915 : and Sen, Trans. (Jhem. Soc. vol. cxv. pp. 467-8, 
1919) that the finer sol is less stable. In 1915 one of 
us pointed out the obvious connexion with the increased 
facilities of coalescence. The smaller particles have a 
more vigorous Brownian movement due to the smaller 
frictional resistance of the medium. This would be clear 
from the well-known equation of Einstein. The diminution 
in the mean distance between the particles also increases 
the rate of collisions. It w r as stated that the adsorption 
theory does not take these factors into consideration. 
Recently Smoluchow T ski (Zeit. Phys. Chem. vol. xcii. 
p. 129, 1917) has been able to formulate the progress 
of the coalescence with time. His attention was drawn 
to the subject by Zsigmondy. Bredig (Anorganische 
Fermente, 1901, p. 15) suggested as the cause of coalescence 
an increase in surface tension with a decrease in the 
electric density on the particles. Zsigmondy (Zeitsch. 
Physikal. Chem. vol. xcii. p. 500, 1918) modified this idea 
in the sense that there is an attraction, between the particles 
which increases with decrease in the electric charge. As 
a result of this attraction he assumes that when one particle 
comes within a certain distance of another, the two coalesce. 
This distance is taken as a measure of the force of attraction 
and is called the radius of the sphere of action. It has 
been shown by Zsigmondy that the time required for a 
definite colour-change in a gold sol gradually decreases 
with rise in electrolyte concentration till it reaches a 
minimum t, which does not change any further with higher 

* Communicated by Prof. F. G. Donnan, F.R.S. 

t Similar minimum times have been observed with cupric sulphide 
and mercuric sulphide sols by the writers. A copper sulphide sol gave 
two minutes as the time necessary for the appearance of visible clots 
when the concentration of the precipitating electrolyte (barium chloride; 
was varied from N/300 to N/20. At dilutions higher than N/300 the 
time was observed to increase as usual (Mukherjee and Sen, loc. cit.). 

Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. X 



306 Profs. J. N. Mukherjee and B. C. Papaconstantinou on 

concentrations. This was assumed to prove that the radius 
of attraction reached a maximum value. 

Smoluchowslu utilized this idea of a sphere of action 
to avoid a consideration of the forces that influence the 
coalescence. He considers the probability of particles 
coming within their mutual sphere of action when the 
radius of the sphere has a constant value determined by 
the conditions. It is assumed that as soon as a particle 
comes within the sphere of attraction by virtue of its 
Brownian movement the two particles coalesce. This dis- 
continuous view of the obviously continuous process of 
coalescence was assumed to avoid a consideration of the 
nature and distribution of the forces that are present. 

Considering the effect of the motion of each particle 
and also that each of the aggregates acts as a condensation 
centre, he derives the following equations : 

2» = ^p , (l) 

^=~°TV 2 , (2) 



K) 



(a.n .t) k l ". 

^.(S^" • .... (3) 

where ' ; w " denotes the total number of particles originally 
present per unit volume before coalescence begins. They 
are all assumed to be spherical and equal in size. u t '•' is 
the time in seconds that has elapsed since the electrolyte 
and the sol have been mixed. " T " is a constant charac- 
teristic of the rate of coagulation and is given by 

T = ^Tr.D.Ka.V ^ 

where "D"* is the diffusion constant as given by Einstein's 
equation; a = 4.7r.D.Ra, and Ra is the radius of the 
sphere of action. 

TT 9 1 

* 1) = ^ =77— — ■■- , where 11= the gas constant, 

1 ° )7r • V •> S = the absolute temperature, 

N =Avogadro's number, 
»7 = the viscosity, 
and r= radius of the pavticle. 



the Kinetics of the Process of Coagulation. 307 

%ti denotes the total number of particles in all stages 
of coalescence in unit volume when the time is " t " ; 
n l denotes the number of the primary particles whose 
original number was n Q at the time " t " ; n k denotes the 
number of particles of the ki\\ stage of coalescence — that is, 
the number of aggregates each of which consists of u k " 
of the primary particles. " k" is evidently an integer. 
In 1918 Zsig.nondy published the results of an investi- 
gation to test this theory. He restricted his investigation 
to the rate of decrease in the primary particles (green in 
the ultramicroscope) in a colloidal gold sol when the 
minimum time of coagulation has been reached. He found 
that Ra = 2*2 times r, the radius of the particles. Similar 
values were obtained by Westgren and Reitstotter (Zeitschr. 
Phijs. Chem. vol. xcii. p. 600, 1918) with more coarsely 
dispersed gold sols. The value of Ra/r, however, varied 
in one experiment from 1'4 to 3*8. The recent experiments 
of Kruvt and Van Arkel (Rec. Trav. Chim. Pays-Bas, 
vol. xxsisc. [4] p. 056, vol. xl. p. 169, 1920) show greater 
variations. They are of opinion that there is some regularity 
in these variations. They could not observe a maximum 
value of Ra/r equal to 2. They found a maximum value 
equal to 0'73. 

iSmoluchowski, assuming from the data of Zsigmondy 
available at that time that Ra/r = 2, points out that the 
maximum rate of coagulation is reached when each collision 
between two par tides is successful in bringing about a 
coalescence. When the rate of coagulation is slower, 
all the collisions are not successful in bringing about a 
coalescence of the particles. If " e " is the fraction of the 
collisions that are successful in bringing about coalescence, 
then " T ,J in equations (1) and (2) takes the form 

T _3 Np.iy r , 

4Ra,rco.0.e' ' [ ° J 

where !N" , Ra, n , 6, and rj have the same meaning as in 
equations (1) and (2). 



Putting 



we nave 



^^v__l (6) 

n n 



t l + /3.e.t 



(?) 



X 2 



308 Profs. J. N. Mukherjee and 13. 0. Papaconstantinou on 

Since only " e " is variable, a comparison of the coagu- 
lation time " t " for the same change in the sol makes it 
possible to determine the variation in the percentage of 
successful collisions and its dependence on the conditions 
of experiment. When the maximum rate is reached, e = l 
and hence a measure of the absolute value of e is possible. 

Problems awaiting solution. — A. glance through the experi- 
mental work would show that the assumption of the constancy 
of "T" is not well justified. The simplicity of Smolu- 
cbowski's equations consist in that there is only one constant. 
The experimental limitations are great, and it is quite possible 
tbat the discrepancies are due to the defects of the ultra- 
microscopic method. The other possibility is that the 
simplifying assumptions of Smoluchowski — for example, 
the constancy of "T" independent of the stage of co^ 
alescence — are not true within narrow limits. Tt is of 
great interest to know the limits within which these 
equations are valid. 

The important questions that await solution in this 
connexion are : 

{a) the limits within which the above equations are 
valid ; and 

(b) if the above equations are valid, the variation of e 

with concentration of electrolyte ; 

(c) the dependence of e on the electric charge ; 

(d) the variation of e with temperature. 

In the following an account of an attempt to examine 
these factors, with the exception of (c) , is recorded. 

Indirect Methods. — Variations in physical properties that 
occur simultaneously with the process of coagulation can be 
utilized to measure the rate of coalescence. 

Smoluchowski pointed out that the viscosity measure- 
ments of Gann (Koll. Chem. Beihefte, vol.viii. p. 67 (1916)) 
do not satisfy the main requirements of his equations — 
namely, a similarity in the form of the curves (showing the 
variation in viscosity with time) independent of the nature 
of the electrolyte. He concludes that viscosity changes do 
not form a measure of the coagulation process. Yet he 
considers that the method is suitable for a quantitative 
comparison of the effect of various concentrations on the 
values of e when the curves are similar. 

The variation in physical properties, however, is likely 






the Kinetics of the Process of Coagulation. 309 

to show the validity of the fundamental equations of 
Smoluchowski. The fact that the curves showing the 
change in viscosity with time are dissimilar shows that 
these assumptions are not justified, and Smoluchowski 
thinks that " T " is dependent on the magnitude of the 
aggregates. 

Since as yet it is not possible to express physical pro- 
perties — e. [/., the viscosity or the absorption of light — in 
terms of definite functions of the number and size of 
particles, a quantitative comparison of different sols is 
not possible by indirect methods. We have, therefore, to 
restrict ourselves to the same sol. 

Experiments icitJi Gold Sols. — An examination of the 
changes in the colour of gold sols on the addition of an 
electrolyte showed (Mukherjee and Papaconstantinou, Trans. 
Chetn. Soc. vol. cxvii. p. 1563 (1920)) that the variation in 
the absorption of light of gold sols affords an easy and 
accurate method suitable for this purpose. The gold sols 
prepared by the nucleus method of Zsigmondy conform 
very nearly to the requirements of equations (1) to (3) 
in so far as the particles are fairly uniform in size. It 
would be very convenient to work with a sol with re- 
producible properties, as data obtained on different dates 
with different preparations could be rigorously compared. 

It was found that a sol on standing for some time under- 
goes somewhat irregular changes, which may in part be due 
to dust particles getting in accidentally. In spite of all 
precautions, one cannot be sure that there is no such variation 
in a particular sample. This variation is not wholly due to 
the fungus that grows in these sols. For this reason it is 
necessary to vary one factor only at a time and compare 
its effects. 

The comparison was therefore restricted to the same sol 
so long as it showed no variation in its properties. 

The Constancy of^T" in Equations (1) to (3) during 
the Process of Coalescence. 

According to the simple assumptions of Smoluchowski, 
the progress of coalescence should be uniformly the same 
for various electrolytes and for their different concentrations. 
The constancy of u T " implies that if we assume a series of 
consecutive stages of coalescence of a sol — under a definite 
set of conditions, namely a definite electrolyte concentration 



310 Profs. J. N. Mukherjee and B. C. Papaconstantinou on 

and temperature — following each other by intervals of time 
equal to "Si," they are each characterized by a definite 
number and manner of distribution of particles of each 
category (primary, secondary, etc.). Let us indicate the 
stage of coalescence corresponding to the time " t " seconds 
(since the sol and the electrolyte were mixed) under the 
given conditions by the numbers 

2N, N x , N„ N ? „ . . . N* . . . , 

where the subscripts refer to the number of primary particles 
by the union of which the aggregate is composed. Thus 
Nfc denotes the number of aggregates, each of which is 
composed of "&" primary particles. "A:" is evidently 
an integer. 

Similarly let us denote the stage of coalescence corre- 
sponding to the time t' ( = t + At) by 

, 2N', N,', i\Y, N 3 ', ... K,/ ... . 

These stages of coalescence are independent of external 
conditions so long as equations (1) to (3) are valid. The 
only change that external conditions can bring about is 
a variation in the value of T — that is, if the external 
conditions are varied the sol will always pass through the 
same consecutive stages of coalescence and only the rapidity 
of succession of these stages will be determined by them. 
Any property which varies continuously with the progress of 
coalescence without having any maxima or minima can be 
utilized to characterize the stages of coalescence ; for each 
value of this property is characteristic of the time that has 
passed since the mixing of electrolyte and sol. According to 
the equations of Smoluchowski, the times taken to reach any 
particular stage depend only on the value of " T/' which is 
constant under a definite set of conditions. Let us compare 
two different electrolytes, A and B, of concentrations C\ 
and C 2 . Let us suppose that after the time " t " the stage 
of coalescence indicated by 

2 N, N,, N 2 , N 3 , . . . N t . . . 

has been reached when the electrolyte is u A " of con- 
centration Ci. This stage of coalescence has a definite 
value for the physical property we are considering, and 
is independent of the value of T. Let us assum£ that " T l *'■ 
and '"T 2 " are the corresponding values of "T" for the two 
cases. To be definite, we shall consider the variation in the 



the Kinetics of the Process of Coagulation. 311 

total number of particles of all categories, which varies 
continually with the progress of coalescence. Let us 
assume that at the times " ^ " and " £ 2 " both electrolytes 
have reached a state at which the total number of particles 
is the same. From equation (1) we have, therefore, 



l+rp 1 + rlT 

or 

t "~ r r * 



(9) 



The general equation (3) may be written as 



n 



(i + i) 



<\* +i 



(10) 



Since n and k are constants, if ^ is constant, n k has a 

fixed value — that is, the condition -J- =J- , which is deduced 

l-i 1-2 

from the condition that ^n has a fixed value, also implies 
that the values of n u n 2 , ft3, ... n^ are the same in both cases. 
This means, in other words, that a definite value of %n fixes 
unequivocally the stage of coalescence. Therefore, from 
the deduction that the successive stages of coalescence are 
always the same and depend only on the time, any property 
of the sol that varies continuously can be utilized to re- 
present a fixed value of %n or n x or a definite stage of 
coalescence. A definite value of this property is thus 
characteristic of the stage of coalescence. It also follows 
from the above considerations that all curves showing a 
variation of this property with time should be similar. A 
deviation from this similarity, in itself, would mean that 
equations (1) to (3) do not represent the facts. 

The absorption coefficients of gold sols for different wave- 
lengths change on addition of an electrolyte in a complex 
manner. The theories of the colour of these sols as 
advanced by Maxwell Grarnett (Phil. Trans, vol. cciii. A, 
p. 385, 1904; vol. ccv. A, p. 237, 1906) and by Mie 
(Ann. der Phys. [4] vol. xxv. p. 377) would lead one 
to expect that any change in the number and manner of 



3 1 2 Profs. J. N. Mukherjee and B. C. Papacon stun tin ou on 

distribution of the particles n 1? n 2 , etc. will produce a 
great change in the optical properties of the sol. This 
is in agreement with observations. Now, if the successive 
stages of coalescence were independent of the nature and 
concentration of the electrolyte, then the manner of 
variation of the complex absorption would be the same 
in each case. The absorption in the red region of the 
spectrum varies continuously, corresponding to each value 
of the absorption coefficient for a particular wave-length in 
this region ; the values in the other parts should be fixed. 
If the contrary holds good, then the conclusion is obvious 
that the successive stages of coalescence are not inde- 
pendent of the nature and concentration of the electrolyte 
as assumed by Smoluchowski. 

It has been found that for the stage indicated by the 
value of the coefficient of absorption for 683 /£/u = 0*4985, 
the values of the coefficient for the other wave-lengths 
given in the following table in column II. are independent 
of the nature of the electrolyte. 

The concentrations of the electrolytes were such as to 
produce rapid coagulation. In columns III. and IV. the 
coefficients of the "nucleus sol" have been given for 
the original sol and for the stage of coagulation cha- 
racterized by the value of the coefficient for 683 /xyu, = 0'4156 
(Mukherjee and Papaconstantinou, loc. cit.). 



Wave-length, 



Table I. 

Absorption coefficients (Jc). 



iafifi. I. II. III. IV. 

683 0-0453 0-4985 0*0376 0-4156 

602 0-1055 0-3679 0-1131 0409 

583 0-1518 0-3388 0-1595 0-3986 

563 0-2076 0-3294 0-2076 0-336 

547 0-2512 3238 0-2867 0-3732 

523 0-3780 0-3780 0-3780 0-3882 

506 0-4647 0-3581 0-3882 0-3780 

475 0-3581 03198 0-3581 03780 



Comparison of the Values of " T " as a Test of 
Smoluchowski' 's Theory. 

Since the absorption coefficient in the red region varies 
continuously with the coagulation and its magnitude is 



the kinetics of t lie Process of Coagulation. 



313 



sufficiently great, a definite value of the abso rption co- 
efficient for a fixed wave-length (683 /jl/jl) can be taken 
as representing a definite stage of the coalescence. 

In the following tables the absorption coefficients at 
different times are given for the wave-length 683 fi/ju. 
The tables are taken from the paper by Mukherjee and 
Papaconstantinou, loc. cit. 



Table II. 
Electrolyte : Potassium Chloride. 



Time in minutes, 

after mixing 

equal volumes of ( — 

electrolytes and sol. N/24. 
0-0453 

05 03732 

1 0-438 

1-5 0-4497 

2 

3 — 

5 - 



13 — 

15 — 



Absorption coefficients 
for various concentrations 



N/26. 


N/28. 


0-0453 


0-0453 


0-2867 


01683 


0-3630 


02257 


0-4046 


— 


0-438 


0-2777 


0-4497 


0-3431 


— 


0-3836 


— 


0-4263 


— 


0-438 


— 


0-4497 



Table III. 
Potassium Nitrate. 



Concentrations. 



Times. 



N/24. N/26. N/30. 

— 00453 00453 0-0453 

0-5 0-3336 — — 

1 0-4263 0-2866 0269 

1-5 0-4497 0-3271 03143 

2 0-3629 0-3336 

3 0-4156 0-3732 

4 0-438 0-394 

5 0-4497 — 

8 — — 0-4263 

10 — — 0-438 

16 — — 0-4497 



314 Profs. J. N. Mukherjee and B. C. Papaconstantinou on 

Table I V. 
Barium Chloride. 

Concentrations. 

Times. * *• -. 



0-852N/9OO. 0-852N/1000. 0-852N/1100. 

— 00453 0-0453 0-0453 

1 0-2257 — 0-1603 

2 0-2867 — 0-2007 

4 0-3529 — 0-2687 

5 0-3836 0-3051 0-3051 

7 0-438 0-3431 0-3237 

8 0-4497 0-3336 

9 0-4497 03836 0-3529 

11 — 0-4263 0-363 

13 — 0-4497 0-363 

16 — 0-3732 

The limits within which the rate of coalescence could be 
varied were restricted by the fact that when the rate is 
slow the particles begin to settle, leaving a clear layer 
at the top, and the measurements are not reliable. Also, 
with time, some of the particles stick to the sides of the 
vessel. Lastly, it is difficult to avoid dust particles for 
a long time. 

The values given in Tables II.- V. were plotted graphi- 
cally, and the time intervals given in Tables V.-VI1. below 
were determined from these curves. 

Each of these curves is characterized by a definite value 
of T (or e). Corresponding to the three concentrations of 
any one of these electrolytes, there are three intervals which 
must pass in order that the absorption coefficient may have 
the same value. These intervals are co-related by the 
following relation according to equations (1) to (3) 
or (6) :— 



h _ ^2 % 

T, ~ T, ~ IV 



(11) 



1 x 2 



or 



^:t 2 :^ 3 = T 1 .T 2 :T 3 = i:i:-. .... (12) 
e t e 2 e 3 

Since T 1? T 2 , and T 3 are constant, the ratio of the time- 
parameters corresponding to the same absorption coefficient 
should be independent of the absolute value of the absorp- 
tion coefficient. Corresponding to different values of the 
absorption coefficient we get different values of t l9 t 2i and t 3 . 
All these values should show a constant ratio. In the 



the Kinetics of the Process of Coagulation. 

following three tables this comparison is made for the 
electrolytes mentioned in Tables II.— IV. 

Table V. 



315 
three 



Values of 
ibsorptiou 
coefficient. 



0370 
0-400 
0-438 
0-445 



Absorption 
coefficients. 



0-350 



0-400 
0-425 
0-445 



Absorption 
coefficients. 



0-300 
0-327 
0350 
0370 



Electrolyte : Potassium Chloride 

Time in seconds. 



N/24. 


N/26. 


N/2S. 


t v 


t.,. 


*r 


27 


65 


255 


35-40 


85 


345 


60 


120 


780 


75 


180 


900 



Ratios. 



Average 

Extreme deviation from average 

Table VI. 

Electrolyte : Potassium Nitrate. 
Tim<?s, 



-"-1 


: T., 


: T,. 




: 2-47 


: 9-44 




: 23 


: 93 




2-0 


: 13-0 




: 2-4 


: 12-0 


1 


: 2-3 


: 11-4 



7-4 % 19 % 



N/24. 

or. -i 

1 07 

30/^' 


N/26. 
105 




N/28. 

h. 

145 


1 


Ratios, 

T 2 : 

: 39 : 


T 3 . 

5-4 


45 


165 




255 


1 


3-7 : 


5 - 7 


60 


210 




375 


1 


3-5 : 


6-2 


90 


300 




780 


1 : 


3-3 : 


8-6 




Average 




1 


: 36 : 


6-3 


sme variation from 


average 




8-8% 


33% 



Table VII. 
Electrolyte : Barium Chloride. 

Times. 



N/24. 


N/26. 


N/28. 




** 


t,. 


k 


T i 


135 


277 


345 


1 


165 


345 


430 


1 


225 


430 


540 


1 


265 


480 


780 


1 



Ratios 

T 2 
2 

2-1 
1-91 
1-81 



Average 1 : 1'93 : 264 

Extreme variation from average 6 % 13 % 

It will be seen from Tables V. to VII. that the agreement 
is as good as can be expected. The variation in T is as great 
as 11 times, but the ratios are constant. The agreement 



316 Profs. J. N. Mukherjee and B.C. Papaconstantinou on 

shows that the ratios of the values of T are independent of 
the time or the stage of coalescence. The ultramicroscopic 
measurements so far made show even during one experiment 
a much greater variation in T, as will be evident from the 
following tables : — 

Table VIII. (a). 
(Observer : Zsigmondy.) 

Values of /3'= rTl . 



Series D. 


Series E. 


Series F, 


0-083 


0-105 


0-040 


0028 


0-058 


0-0195 


0-0302 


0-049 


0-0183 


00309 


0-0475 


0-0153 


— 


0-0403 


0-0187 


— 


— 


00126 



Zsigmondy used high concentrations of electrolyte for 
securing a rapid rate. When the rate of coagulation is 
slow and the duration of experiment is greater than a 
few minutes, he found that impossible values of /3' are 
obtained. He thinks that the presence of impurities in 
the water used in diluting the sol for ultramicroscopic 
observations is the cause of this irregularity. In his case 
■the m iximuin time covered by the experiments is 80 sees. 
Similarly, Westgren and Reitstotter, working with coarse 
gold sols, find the following range of variation in the 
constant : — 

Table VIII, (6). 
(Observers : Westgren and Reitstotter.) 
Ra 







Values of — 


-* 






Series I. 


Series II. 


Series III. 


Series IV. 




3-74 


2-56 


2-75 


3-41 




2-47 


2-81 


2-60 


2-80 




2-07 


2-33 


2-17 


2-60 




2-10 


2-31 


2-40 


2-48 




2-09 


2-31 


212 


2-14 




1-62 


— 


— 


— 




1-41 


2-16 


215 


215 




— 


2'19 


— 


205 


Average 


. 2-2 


2-38 


2-36 


2-19 


Extreme variation., 


.. 75% 


10% 


17% 


55% 



the Kinetics of the Process of Coagulation. 317 

Kruyt and Arkel *, working with selenium sol and very 
slow rate of coagulation, find extremely wide variations in T 
in the same experiment. 



Table IX. 

(Observers : Kruyt and Arkel.) 

Values o£ T (in hours) . 



I. 


II. 


III. 


IV. 


2-8 


200 


131 


1-3 


51 


390 


55 


34 


44 


270 


52 


2'2 


(43) 


320 


54 


43 


(157) 


600 


68 


105 


200 


- 370 


55 


40 


— 


510 


— 


— 


— 


440 


48 


37 


— 


— 


— 


52 


__ 


_ 


— 


38 



The above few instances will suffice to show the range of 
variations in u T " during the course of one experiment that 
has been observed in the ultramicroscopic measurements. 

Considering that in Tables Y. to VII. the ratios between 
the different values of T are taken, the range of variation 
is extremely small. The actual deviations in the value of T 
in any one experiment must be much less than the extreme 
variations given. This comparison leaves no room for doubt 
that " T " is a constant in the case of gold sols and within 
the limits of the rate of coagulation that have been studied. 
In fact, these data constitute the best evidence so far recorded 
in favour of the theory of Smoluchowski. 

The Dependence of € on the Concentration. 

Tables V. to VII. show clearly how rapidly e, the 
percentage of successful collisions, increases with con- 
centration. A change of concentration in the ratio 24 
to 28 increases the rate in the ratio 1 : 11 or 1:6 as the 
case may be. It would be extremely interesting to work 
with a sol which is less susceptible to impurities than these 
gold sols. 

* Fee. Trav. CLim. Pays-Bas, vol. xxxix. T4] p. 6o(i (1920) ; [4] 
vol. xl. p. 169(1921). 



318 Profs. J. N. Mukberjee and B. 0. Papaconstantinou on 
Variation o/Tore with Temperature. 

Similarly, by determining the times required to produce a 
definite change in the colour o£ tbe sol for the same electrolyte 
concentration but different temperatures, we can determine 
the variation in e with temperature. 

From equation (3), 

we get 

2 9 - = l+£V.«.« (13) 

Since a definite change of colour is being used, ^ is 
constant, or n 

1 + • € f t = k^, a constant. , . . (11) 

Substituting the value of j3 in (14), we get 

4 Ra . . n . ,^„ 

Since Ra, N , and n are constants, we have 

— — — = k' a constant (16) 

The viscosity of colloidal gold solutions has been found to 
be practically equal to that of water, and the variation with 
temperature can be assumed to be equal to that of water. 
For different temperatures Ave have 

£j . 0! . e l _ t 2 . 6. 2 . 6 2 - 

Vi V2 

Since t x is experimentally determined and and 7) are 
known, variations in e can be compared. 

The experimental data are given below. They are taken 
from the same paper (pp. 1570-71). 



Table X. 



Temperatures. 



Electrolyte. Standards*. 15°. 30°. 50°. 

N/30 Potassium chloride .. V Sol. C. 5 inin. 10 -in in. 8 min. 30 sec. 

N/30 Potassium sulphate . ,, „ D. 30 sec. 10 sec. 10 sec. 

N/30 Potassium nitrate .. „ „ D. 42 ,. 18 „ 12 ,, 



* These i*efer to the protected gold sols used as standards for 
comparison of colour. See loc. cit. 



the Kinetics of the Process of Coagulation. 319 

Table XI. 
Electrolyte: Barium Chloride. Sol. E. 



Con- 
centrations. Standards. 






Temperatures. 




15°. 


30°. 


40°. 


50°. 


0-852 N/1000 


v 2 


7 min. 


6 min 


4 min. 50 sec. 


4 min. 20 sec 


0-852 N/1000 


B 2 


34 „ 


23 „ 


— 


13 „ 30 „ 


0852 N/1200 


V 2 


23 „ 


13 ., 


12 min. 30 sec. 


6 „ 15 „ 


0-852 N/1200 


B 2 


124 „ 


74 „ 


62 min. 


— 



Con- 



Table XII. 
Electrolyte : Strontium Nitrate. Sol. F. 

Temperatures. 



centrations. Standards. 15°. 30°. 50°. 

N/1000 V 3 1 min. 10 sec. 20 sec. 8 sec. 

N/1000 B 3 8 „ 15 „ 1 min. 40 sec. 45 „ 

At 15°, 30°, 10°, and 50°, v /0 has the values 3"96 x 10~ 5 , 
3-31 xlO- 5 , 2'1 x JO" 5 , and 1*7 x 10" 5 respectively. The 
values for the viscosity are taken from the tables in Kaye 
and Labv's hook on Physical and Chemical Constants, 
p. 30, 1919. 

From equation (17) we have 

6 \o° " 6 30° : 6 -l(P ' 6 5U 3 

= (v/tO)ro° : (v/tO'hoo : (r}/t6)^ : ( V /t0) 5O o. 
Table XIII. 

Temperatures. 
Electrolyte. 

N/30KC1 N//0X1O 7 

N/SOK.SO^ 

N/30KNO 3 



15°. 


30°. 


50°. 


1-32 


0-50 


0-33 


3-2 


33 


17 


943 


18-4 


14-0 



Table XIV. 
Electrolyte : Barium Chloride. 



Con- 
centrations. Standards. 
0-852 N/1000 V 2 77/^ XlO 8 

b; 

Ratio between f V., 

v/to 1 b; 



Tempera ti 



0852 N/1200 V 2 rj/tOxW 

B 2 

Ratio between J V a 

,,-te [ B." 



15°. 
9 43 
1-94 


30°. 
92 
24 


40°. 
7-24 


50°. 
6-54 
2-1 


100 

100 


: 98 
: 123 


77 


: 70 
: 108 


2-87 
53-22 


421 

74-:> 


2-8 
04 


45 


100 
100 


: 148 
: 140 


: 98 
: 106 


: 158 



320 The Kinetics of the Process of Coagulation. 

Table XV. 
Electrolyte : Strontium Nitrate. 

Temperatures. 
Concentrations. Standards. '77^ ^ 50°~^ 

N/1000 Y 3 77/^xlO 6 '565 1-65 2-1 

B 3 ri/tBxlO 1 '80 3"3 377 

Ratio between f V 3 100 : 291 : 371 

nltB 1 B 3 ... 100 : 410 : 430 

Since 7] = t6 is a constant for a definite electrolyte con- 
centration and temperature according to Smoluchowski's 
equation, the ratios should be independent of the standard 
used. This is true within the limits of experimental error 
with -852 N/1200 Barium Chloride. In the other two cases 
the variations are not great considering that we are com- 
paring the ratios. A slight variation in each value will be> 
magnified in the ratio. Taking into account the probable 
experimental error, it can be said that e is roughly constant 
in each experiment. 

On the other hand, the variation in e with temperature is 
considerable. We have already seen that the irregularity 
in the variation of e means that the precipitating power of 
the ions changes with the temperature (Mukherjee, Trans. 
Chem. Soc. vol. cxvii. p. 358, 1920). 

Further experiments with arsenious sulphide are in 
progress on similar lines. 

Summary. 

(1) It has been shown that the equations of Smoluchowski 
on the rate of coalescence of the particles of gold sols agree 
with the results obtained by the writers. 

(2) It has been suggested that the disagreement of the 
ultramicroscopic measurements with this theory may in part 
be due to the difficulties inherent in them. 

Our best thanks are due to Professor F. Gr. Donnan for 
his kind interest and encouragement, and also to our friend, 
Professor J. 0. Ghosh. 

Physical Chemistry Department, 

University College, London. 



[ bsi ] 



XXIX. The Adsorption of Jons. By Jnanendra Nath 
MuKHERJEE, D.Sc, Professor of Physical Chemistry in 
the University of Calcutta*. 

IN a paper in the Transactions of the Faraday Society 
(Far. Soc. Disc. Oct. 1921) an attempt has been made to 
define the nature of the adsorption of ions to which the 
origin and the neutralization of: the charge oE a colloidal 
particle are due. The origin of the charge was assumed to 
be due to the adsorption of ions by the atoms in the surface 
as a result of their chemical affinity. 

It was pointed out that the adsorption of one kind of ions 
will impart a charge to the surface, in virtue of which ions 
of opposite sign will be drawn near the surface. In the 
liquid there remains an equivalent amount of ions. of opposite 
sign. The electrical energy will be a minimum when these 
ious are held near the surface so that the distance between 
the oppositely charged ions has the minimum value possible 
under the conditions, and they will be held opposite to the 
ions chemically adsorbed. An "ion'' so held will not be 
" free" to move if its kinetic energy is less than " W " the 
energy required to separate the ion from the oppositely 
charged surface. The number of such "bound " ions deter- 
mines the diminution in the charge of the surface. When 
the concentration of ions of opposite charge in the liquid is 
small the number of ions "held" to the surface by electrical 
attraction will be small. 

If the chemically adsorbed ions have a valency equal to 
" N l5 " and if " N 2 " is the valency of the oppositely charged 
ions in the liquid in contact with the surface, then 

Ni.Xs.E 2 
vv ~ D.a ' w 

where E = the electronic charge, x = the distance between 
the centres of the ions at the position of minimum distance, 
and " D " is the dielectric constant of water. 

Depending on the concentration of the oppositely charged 
ions in the liquid near the surface, at any instant a certain 
number of the " chemically adsorbed " ions are " covered " 
by ions of opposite charge. In the liquid near the surface 
there are always a number of free ions equivalent in amount 
to the " uncovered" chemically adsorbed ions on the surface. 
The total amount of ions of opposite sign both " bound" and 

* Communicated by Prof. F. G. Donnan, F.R.S. 
Phil Mag, Scr. 6. Vol. 44. No. 260. Aug. 1922. Y 



322 Prof. J. N. Mukherjee on 

"free" is equivalent to the amount of ions "chemically, 
adsorbed." These " free" ions form the second sheet of the 
double layer. It is evident that as a result of their thermal 
motion the mean distance between the two layers will be 
greater than "#." 

The charge of the surface was treated as due to discrete 
charged particles widely separated from each other compared 
with molecular dimensions. It was shown in the previous 
paper that this view gives a rational explanation of the fact 
that a reversal of the charge of a surface can be brought out 
only by polyvalent ions of opposite charge. 

The equilibrium conditions were discussed and the equa- 
tions deduced were shown to be in agreement with the 
valency rule, the influence of the mobility of the oppositely 
charged ion, and with the influence of concentration on the 
charge o£ the surface. Only the theoretically simplest case 
has been discussed in the earlier paper. In the present paper 
the more important facts connected with the adsorption of 
ions are discussed from this point of view, and it will be 
seen that this view gives a simple explanation of most of 
the general conclusions already arrived at on experimental 
grounds. 

1 heories of Adsorption. 

Before proceeding to discuss the adsorption of ions it 
will be convenient to deal briefly with the different views 
advanced to account for adsorption in general. The with- 
drawal -of a solute from a solution by a solid may be the 
result of the formation, of definite chemical compounds, of 
solid solutions, of mixed crystals and surface-condensation. 
In many cases all these changes are simultaneously present. 
In this paper the word " adsorption " denotes condensation 
or combination, at the surface only, without the interpenetra- 
tion of the adsorbed substance throughout the mass of the 
adsorbent (Mecklenburg's criterion, Z. Phys. Chem. Ixxxiii. 
p. 609 (1913) ; cp. also the sense in which the term is used 
in deriving Gibbs's equation). 

Faraday (Phil. Trans, cxiv. p. 55 (1834)) in his well-known 
explanation of the catalytic combination of hydrogen and 
oxygen on platinum surfaces,, remarks "that they are de- 
pendent upon the natural condition of gaseous elasticity 
combined with the exertion of that attractive force, possessed 
by many bodies, especially those which are solid, in an 
eminent degree, and probably belonging to all, by which they 
are drawn into association more or less close, without at the 
same time undergoing chemical combination though often 



the Adsorption of Ions. 323 

assuming the condition of adhesion, and which occasionally 
leads under very favourable circumstances, as in the present 
instance, to the combination of bodies simultaneously sub- 
jected to this attraction." It is remarked further u that the 
sphere of action of particles extends beyond those other 
particles with which they are immediately and evidently in 
union, and in many cases produces effects rising into con- 
siderable importance.''' These remarks of Faraday mean, in 
modern terminology, that there is a sort of combination at 
the surface and that the transitional layer is more than one 
molecule thick. The subsequent views are in a way develop- 
ments of this conception. 

Gribbs treated adsorption from the standpoint of thermo- 
dynamics. A number of important investigations has been 
carried on by Milner (Phil. Mag. [6] xiii. p. 96 (1907)), 
Lewis (PhiL Mag. [6] xv. p. 506 (1908)), ibid. xvii. 
p. -466 (1909)), and Donnan and Barker (Prpc. Roy. Soc. 
Ixxxv. A. p. 552 (1911)). The present position is that the 
amount adsorbed is often considerably greater than what 
could be expected from Gibbs's equation. 

J. J. Thomson (' Applications of Dynamics to Physics and 
Chemistry ') showed that it follows from Laplace's theory of 
capillarity that in the surface layer between two liquids, 
chemical actions may take place which are absent in the 
bulk of the liquids. 

Lagergren (Biliang K. Svenska Vet. Hand. xxiv. p. 11, 
Xo. 115 (1898)) considers that adsorption in the surface of 
solids in contact with aqueous solutions is due to the com- 
pressed state of the water in the surface layer. 

On the experimental side the work of Freundlich and his 
collaborators — [Kapillar-Chemie, 1909 ; Z. Phys. Cheni. 
lix. p. 284 (1907); Ixvii. p. 538 (1909) ; Ixxiii. p. 399 (1910) ; 
Ixxxiii. p. 97 (1913) ; Ixxxv. p. 398 (1913) ; xc. p. 681 (1915) ; 
Koll.-Ch.em. Beihefte,\i. p. "297 (1914) : see also Schmidt, 
Z. Phys. Chem. lxxiv. p. 689 (1910) ; lxxvii. p. 641 (1911) : 
lxxviii. p. 667 (1912); Ixxxiii. p. 674 (1913); xci. p. 103 
(1916). In the last-mentioned paper Schmidt and Hinteler 
conclude that Freundlieh ; s equation represents their experi- 
mental data better than that of Schmidt] — and of others, have 
shown that adsorption-equilibria can be generally expressed 
in terms of the well-known equation of Freundlich : 

xt)ll = a.C l P (2) 

Freundlich expressed the opinion that adsorption is mainly 
due to a decrease in surface tension as suggested by Gibbs, 

Y2 



324 Prof. J. N. Mukherjee on 

In the case of adsorption of gases by solids, Arrhenius 
(Medd.f. k. Vef. Nobelinstitut, ii. N. 7 (1911); Theories of 
Solution, 1912, pp. 55-71) has drawn attention to the 
parallelism between the van der Waals's coefficient "a" for 
the different gases and the amounts of these gases adsorbed 
by charcoal, and he believes that this is definite evidence of 
the compressed state of the surface layer. At the same time 
he lays stress on the chemical aspect — namely, that in 
addition to the attractions between the molecules of the gas 
in the surface layer, one has to consider the chemical 
attraction of the surface atoms and the molecules of the gas. 

Recently, Williams (Proc. Roy. Soc. xcvi. A. p. 287 (1919) ; 
xcviii. A. p. 223 (1920); also Trans. Far. Soc. x. p. 155 
(1914), in which complete references to the literature on 
negative adsorption are given) has treated adsorption from 
the points of view of Lagergren and of Arrhenius in a number 
of interesting communications. 

It may be mentioned here that the disagreement of ob- 
servations with calculations from Gibbs's equation is at least 
in part due to the fact that only one source of change in the 
free energy of the surface layer is taken into account. In 
the simplest case of the interface, liquid-saturated vapour 
(one component system), it is open to objection whether " 7 " 
denotes the total change in free energy of an isothermal and 
reversible-formation of unit surface. Bakker (Z. Phys. Chem. 
lxviii. p. 684 (1910)) has pointed out that if the density of 
the surface layer is different from that of the liquid in bulk 
a second term is necessary to represent the change in free 
energy. 

It is possible that in this particular case this second term is 
negligible in comparison with " 7," the tension per unit length 
at low temberatures, but at high temperatures " 7 ,7 has a low 
value and the saturation pressure is very great, so that the 
second term may be even more important. 

Williams (Proc. Roy. Soc. (Edinburgh), xxxviii. p. 23 
(1917-18)) has drawn attention to the effect of the variation 
of the surface of an adsorbent when adsorbing — a factor 
which is very often neglected. 

Lewis (Z. Phys. Chem. lxxiii. p. 129 (1910) ; also Par- 
tington, c Text-book of Thermodynamics,' p. 473 (1913)) 
has discussed the influence of a variation in the electric 
density on the surface on the form of Gibbs's equation. 

These may be called the physical theories of adsorption. 
The difficulty in accepting them as general theories of ad- 
sorption is that they attempt to explain adsorption in terms 
of a single physical factor j e. g. diminution in surface energy 



the Adsorption of Ions. 325 

or a layer under great internal pressure. The necessity for 
recognizing the existence of a sort of chemical interaction (as 
Arrhenius has suggested) becomes evident when one con- 
siders the specific nature of adsorption processes. This point 
has heen justly emphasized by Bancroft in recent years. 
Besides his papers in the ' Journal of Physical Chemistry,' 
compare 'Applied Colloid Chemistry/ 1921, p. 111). 

The chemical point of view has been put clearly by Lang- 
muir (J. Amer. Chem. Soc. xxxviii. p. 2221 (1916) ; xxxix. 
}). 18-18 (1917)). He believes that adsorption is due to the 
chemical affinities of the surface atoms. Considering the 
thermodynamic equilibrium between molecules of a gas at 
the surface and those in the surrounding gas he has deduced 
the following equations correlating the variation of the ad- 
sorbed amount with its pressure, 

where " di' is the fraction of the solid surface covered and is 
a measure of the amount adsorbed, vi is the rate at which 
the gas would evaporate if unit area of the surface were 
completely covered, "//," is the number of gas molecules 
striking unit area of the surface per second and is given by 

//,= 43*75 X 10~ 6 — — ;f- — — , and "p ,} denotes the pressure of 

the gas, "T" its absolute temperature, and " M" its molecular 
weight. a denotes the fraction of the total number of 
collisions of the molecules of the gas that leads to a condensa- 
tion on the surface; it is usually close to unity and evidently 
can never exceed unity. Some interesting applications of 
his theory to catalysis of gaseous reactions by solid surfaces 
are given, This theory explains many phenomena which are 
otherwise difficult to understand. 

Michaelis and Rona (Bio-Chem. Zeitsch. xcvii. pp. 56, 85 
(1919)) conclude from the investigations of Michaelis and his 
co-workers that the assumption of special forces at the sur- 
face fails to account for the facts and that adsorption is the 
result of chemical affinity. 

I. The Adsorption of a Constituent Ion by a Precipitate. 

The adsorption of ions is different from the adsorption of 
neutral molecules or groups in that it introduces a new 
factor — an electrically charged surface. The variation in 
the electric charge enables us to follow the net effect of the 
adsorption of the two ions, as the electric charge depends 



326 Prof. J. N, Mukherjee on 

only on the total number of ions (of both signs) fixed per 
unit area of the surface. Kataphoretic and electro-endosmotic 
experiments give us a quantitative idea of the relative ad- 
sorption of both ions. 

The electric charge helps to peptize the adsorbent, and a 
qualitative idea of the adsorption of ions can be formed 
from peptization by electrolytes. An insoluble precipitate 
formed by the union of two oppositely charged ions has a 
marked tendency to adsorb its component ions. In many 
cases the connexion between the adsorbed ion and the 
electrical charge has been established. These instances have 
been given in the earlier paper. The nature of the chemical 
forces responsible for this adsorption has also been defined. 
Instances of adsorption of ions as judged from peptization by 
electrolytes are given below. 

Bancroft (Rep.' Brit. Assoc, p. 2 (1918)) remarks: — ;; It 
seems to be a general rule that insoluble electrolytes adsorb 
their own ions markedly, consequently a soluble salt having 
one ion in common with a sparingly soluble electrolyte will 
tend to peptize the latter. Freshly precipitated silver halides 
are peptized by dilute silver nitrate or the corresponding- 
potassium halide, the silver and the halide ions being ad- 
sorbed strongly. Many oxides are peptized by their chlorides 
and nitrates, forming so-called basic salts. Sulphides are 

peptized by hydrogen sulphide The peptization of 

hydrous oxides by caustic alkali can be considered as a case 
of adsorption of a common ion or as the preferential adsorption 
of hydroxy lion. Hydrous chromic oxide gives an apparently 
clear green solution when treated with an excess of caustic 
potash ; but the green oxide can be filtered out completely 
by means of a collodion filter, a colourless solution passing 
through."" 

" Hanztsch considers that hydrous beryllium oxide is 
peptized by caustic alkali, copper oxide is peptized by con- 
centrated alkali, and so is cobalt oxide. In ammoniacal 
copper solutions part of the copper oxide is apparently colloidal 
and part is dissolved. Freshly precipitated zinc oxide is 
peptized by alkali, but the solution is very unstable " (cp. also 
negative hydroxide sols — Freundlich and Leonhardt, Koll. 
Chem. Beihefie, vii. p. 172 (1915)). 

At least in some of these cases the formation of new com- 
plex anions is possible, and it is not definitely known to what 
ion the peptization is due. Regarding the peptization of 
stannic acid gel by small quantities of alkali, Zsigmondy 
(Kolloidchemie, p. 122 et seq. (1920) ; also Varga, Koll. 
Chem. BeiJiefte, xi. p. 26 (1919)) remarks: "Dieses kann 



the Adsorption of Ions. 327 

sowohl auf Adsorption des gebildeten Kaliumstannats wie 
audi daraut' zUruckt'iihren sein, das Kaliumhydrat mit 
den Oberflachenmolekeulen der Zinnsaureprimarteilchen in 
Reaktion tritt, wobei diese von der Oberflache der Primar- 
teilehen festgehalten werden." 

The view suggested bv the writer to account for the ad- 
sorption of a common ion, leads one to expect that ions 
which can displace one of the constituent ions in the crystal 
lattice should also be adsorbed. Marc {Z. Phys. Chem. lxxxi. 
p. 641 (1913)) has observed that crystalline adsorbents adsorb 
crystalloids to any marked degree only when they can form 
mixed crystals with them and are isomorphous with them. 
Paneth and Horrowitz (Physik. Zeitsch. xv. p. 924 (1914)) 
have noticed that oc the radio elements those only will be 
adsorbed that can form insoluble salts with the common 
ion of the adsorbent and can also form mixed crystals with 
the adsorbent. This kind of adsorption is somewhat different 
from the type we have considered for, as Paneth has pointed 
out in his case, an actual interpenetration of the two non- 
common ions is occurring in the crystal lattice. Thus 
radium is taken up by barium sulphate giving out to the 
solution barium ions in exchange. Such an interchange will 
not impart a charge to the surface. 

Attention may also be drawn to the explanation advanced 
by Bradford (Biochem. J. x. p. 169 (1916) ; xi. p. 14 (1917)) 
to account for zonal precipitations, first siudied by Liese- 
gang. Bradford thinks that the adsorption of a constituent 
ion is responsible for their formation. From the numerous 
instances given aboA r e, this conception seems to be quite 
plausible. It is probable that other factors have also an 
influence on the process (Hatschek, Brit. Assoc. Rep. p. 24 
(1918)). 

II. The Variation of the Density of the Electric Charge 
with the Concentration of an Electrolyte. 

In the previous paper the particular case when the charge 
o£ the surface is due to strong chemical adsorption of ions of 
one kind and when the added electrolytes have not any ions,, 
subject to the cheniical affinity of the surface atoms, has been 
fully treated. In this case it was assumed that the number 
or! ions adsorbed at the surface by chemical affinity remains 
constant. The experimental data of Elissafoff on glass 
and quartz agree well with equations deduced from these 
assumptions, on the basis of the theory of electrical 
adsorption. 



328 Prof. J. N. Mukherjee on 

The general case, however, is that : 

(a) At low concentrations the density of the charge on 
the surface at first increases to a maximum and at higher 
concentrations falls gradually towards a null value when the 
oppositely charged ions are monovalent. 

(b) On the other hand, when the oppositely charged ions 
are multivalent or complex organic ions the charge passes 
through a null value, becomes reversed in sign, and ugain 
reaches a second maximum, after which it falls slowly 
(Ellis, Z. Phys. Chem. lxxviii. p. &21 (1911) ; lxxx. p. 597 
(1912) ; lxxxix. p. 115 (1914) ; Powis, Z. Phys. Chem. 
lxxxix., pp. 91, 179 (1914) ; Piety, Compt. Rend. cliv. 
pp. 1411, 1215 (1912); clvi. p. 1368 (1913); Young and 
Neal, J. Phys. Chem. xxi. p. 1 (1917); Krnyt, VersL Ron. 
Akad. v. Wetensch. Amsterdam, 27th Juin, 1914, also Koll- 
Zeitsch. xxii. p. 81 (1918)). 

The usual explanation is as follows: — 
The adsorption-isotherms for the two ions can be written 
as 

^? =«_. A and ^ =«..«"*, ... (3) 

m K m A 

where the subscripts A and K refer to the anion and the 
cation respectively. To explain the increase in the charge 
at low concentrations it has to be assumed that 

a A >a K and fi A </3 K . .... (4) 

Thus in a paper read at the Discussion on Colloids arranged 
by the Faraday and the Physical Societies of London, 
Svedberg remarks : " Now as a rule, it happens that for the 
two ions of a salt both a and /3 have different values, e.g. 

a (cation) < a (anion) 

8 (cation) >/3 (anion)." 

It is clear that the equation of the adsorption-isotherm can 
be reconciled with the first increase in the charge. But two 
objections can be raised against this empirical point of view. 
In the first place, no reason is given why the constants a and 
/3 shall have generally the relative values assumed above for 
the cation and the anion. Secondly, these assumptions can- 
not explain the second maximum charge and the subsequent 
decrease observed with multivalent ions of opposite charge. 
It will now be necessary to assume that 

« A <« K and /3a>/3k, .... (5) 

in direct contradiction to the assumptions already made 



the Adsorption of Jons. 329 

(r/>. (4)). Besides, one cannot get any idea lis to why the 
anion is generally more strongly adsorbed at low con- 
centrations. 

The facts can, however, be explained as follows : — 

The negative charge of surfaces in contact with water is to 
be sought for in the chemical natures of the anions and the 
cations. The simpler electrolytes (excluding dyes and 
complex organic ions) have cations whose chemical behaviour 
can be referred simply to the tendency of the component 
atom (e g. of the alkali and alkaline earth metals) to pass 
into the ionic state. These ions do not form any complex 
ions. They form only one type of compounds that are stable 
in aqueous solutions, namely, electrolytes with the atom 
existing as a positively charged ion through the loss of one 
or more electrons. On the other hand, the anions in general 
form types of compounds other than electrolytes, and also 
form complex ions. It is, therefore, possible to imagine that 
anions are subject to the chemical affinity of the surface 
atoms and that the chemical action on the cations is relatively 
small. Complex cations like those of the basic dyes should, 
for the same reason, be easily adsorbable. This is a well- 
known fact. 

If now, the assumption is made that the chemical affinity 
acting on the anion of the electrolyte added is stronger than 
the electrostatic attraction of the surface on the cation, the 
observed variation of the charge with the concentration of the 
electrolyte is easily accounted for. This case corresponds to 
a strongly marked maximum of a negative charge at a low 
concentration of the electrolyte. 

The initial charge of a surface in contact with pure water 
-can be due either : 

(a) to the strong adsorption of an ion of a minute quantity 

of suitable electrolyte associated with the solid, 
(7>) or to the adsorption of hydroxy! ions from water. 

On the addition of an electrolyte the density of the electric 
charge will increase at low concentrations because of the 
chemical adsorption of the anion. The electrical adsorption 
of the cation is smaller as the chemical adsorption has been 
assumed to be stronger. Besides, the electric charge of the 
surface is also not at its maximum. As the surface becomes 
more and more covered by the anions the rate of adsorption 
das/dc — where " da" is the increase in the amount adsorbed 
per unit surface due to an increase in the concentration 
^ dc" — rapidly decreases. Also, the electric charge repels 
the anions, and those only can strike on it that have sufficient 



330 Prof. J. N. Mukherjee on 

kinetic energy to overcome the potential of the double layer. 
The number of collisions is thus not proportional to the 
concentration but rises more slowly. Near about the point 
where the surface becomes saturated the value of dxjdc will 
be almost zero (cp. the shape of the adsorption-isotherms 
of Freundlich, Arrhenius, and Langmuir). On the other 
hand, the electrical adsorption increases continually with 
the concentration and the increase of the charge. It is- 
apparent that soon a balance will be reached between the 
chemical adsorption of the anion and the electrical adsorption 
of the cation. The minimum charge will correspond to the 
stage when dxjdc for the cation is just equal to dxjdc for the- 
anion. 

Beyond this concentration the charge will decrease rapidly,, 
and when the surface has been saturated with the anion the 
subsequent variation in the charge is simply due to electrical 
adsorption. The reversal of the charge by electrical adsorp- 
tion has been discussed in the earlier paper. It is necessary 
to add that as the electrically adsorbed polyvalent cations 
impart a positive charge to the surface, the adsorption of the 
cation decreases and the electrical adsorption of the anion 
becomes possible. As long as there is a positively charged 
surface the adsorption of the anion will increase more rapidly 
with the concentration than that of the cation. A second 
maximum will thus be reached and a decrease in the charge 
will follow. The electrical adsorption of the anion is small 
because of the smallness of the positive charge and an 
initially existing negatively charged surface. A further 
reversal of the charge is not possible, and, in fact, has never 
been observed. 

III. The Actio?i of Acids and Alkalies. 

The works of Perrin and of others (J. Chim. Phys. ii. 
p. 601 (1904) ; iii. p. 50 (1905) ; Haber and Klemensie- 
wicz, Z. Phys. Chem. Ixvii. p. 385 (1909) ; Cameron and 
Oettinger, Phil. Mag. [vi.] xviii. p. 586 (1909)) have shown 
that hydrogen and hydroxyl ions behave exceptionally in 
that they impart to the surface a charge of the same sign as 
they carry. This behaviour is in contrast to that of the 
other univalent ions. 

Perrin attributes their singular activity to the smallness of 
their radii. In order to explain the presence of these ions, 
in excess, in the surface layer, it is necessary to assume some 
sort of a restraining force acting on them at the surface. 

Haber and Klemensiewicz consider that there is an ad- 
sorbed layer of water in the surface by virtue of which the- 



the Adsorption of Ions. 331 

solid .acts as a sort of combined hydrogen and oxygen 
electrode. They treat the subject from the points of view of 
thermodynamics and Nernst/s theory of electrolytic solution 
tension. It has been pointed out by Freundlich (and 
Elissafoff, Z. Phys. Chan. Ixxix. p. '407 (1912)) that 
hydrogen and hydroxyl ions are not the only ions which 
impart a charge to the surface. In many cases, acids haye 
been observed not to reyerse the charge at all. Many sub- 
stances haye a negative charge in contact with pure water. 
These facts show that selectiye adsorption of hydroxyl ions 
has also to be considered. 

This thermodynamic treatment from the point of yiew of 
Nernst's theory does not attempt to explain electro-endosmosis. 
For this purpose it is necessary to conceive of an electrical 
double layer, of which the layer imparting a charge to the 
surface is fixed relative to the mobile second layer. 

Freundlich, and Freundlich and Rona (Koll. Zeit. xxviii. 
5, p. 240 (1921); Kgl. Preuss. Akad. Wiss. Berlin, 1920, 
p. 397, C. 1920, iii. p. 26) have shown that the potential 
measurements by Haber's method are not in agreement with 
those measured by electro-endosmotic experiments. They 
therefore suggest that there are two distinct drops in 
potential as one passes from the solid to the liquid (glass to 
water). The first drop is wholly in the solid and is probably 
of the nature associaied with the Nernst theory of electrolytic 
solution-tensions. 

The second drop is in the liquid and composes the Helm- 
holtzian double layer which it is necessary to assume to 
explain electro-osmosis and cataphoresis. 

At the same time the characteristic effects of hydrogen 
and hydroxyl ions on neutral substances like barium sulphate, 
silver chloride, naphthalene, etc., point strongly to the 
correctness of Haber's fundamental assumption that the 
explanation is to be sought in the equilibrium between the 
hydrogen and hydroxyl ions in the adsorbed layer of water 
and those in the bulk of the liquid. 

Williams (Proc. Roy. Soc. xcviii. A. <p. 223 (1920)) has 
recently suggested that the layer of water adsorbed on a 
charcoal surface is under great internal pressure (about 
10,000 atmospheres). Applying Planck's equation he shows 
t}iat the effect of this pressure will be to increase the con- 
centration of hydrogen and hydroxyl ions in this layer. 
This increased concentration will setup a diffusion potential. 
He draws attention to the difficulties in accepting this view 
of the origin of the potential difference at the surface. In 
the instances considered by Haber and Perrin, the solid has 



332 Prof. J. N. Mukherjee on 

little or no potential difference in contact with pure water, 
and the considerations developed by Williams are not 
applicable. 

Case 1. — The surface is inert. 

We shall assume that the atoms in the surface do not 
exert any chemical affinity on hydrogen and hydroxyl ions 
as such, or on the dissolved acid (or alkali) with which it may 
be in contact. The adsorbed water molecules behave as a 
solid layer, being held by strong chemical forces (Haber, 
loc. cit. ; Hardy, Proc. Roy. Soc. lxxxiv. B. p. 217 (1911)). 
It is clear that the surface will be neutral in contact with 
pure water. The molecules of water in the adsorbed layer 
are in thermodynamic equilibrium with those in the bulk of 
the liquid. It is reasonable to imagine that a transfer of an 
electron is taking place between the hydrogen atom and the 
hydroxyl group in the water molecules in the surface layer, 
as it does in the molecules in the liquid. That is, the water 
molecules are dissociating into ions at a definite rate. Let 
ki nx" be the number of water molecules (in the adsorbed 
layer) passing into the ionized phase per unit area per 
second. For equilibrium, as many hydrogen and hydroxyl 
ions are uniting to form neutral water molecules. Since the 
adsorbed water molecules behave as a solid layer, recombina- 
tions would take place mostly between adjacent hydrogen 
and hydroxyl ions. The recombination will be extremely 
rapid. It can be assumed that at any instant the number of 
hydrogen or hydroxyl ions actually remaining free in the 
surface will be a negligible fraction of the total number of 
water molecules. 

The neutralization of the ions being formed in the surface 
layer can also be brought about by impinging hydrogen or 
hydroxyl ions present in the liquid. In contact with pure 
water the probability of such collisions is small, for the 
concentration of hydrogen and hydroxyl ions is extremely 
small. Thus neutralization of the ions being formed in the 
surface layer is possible in two ways : 

(1) H s ° + OH, 1 — ^HOH— the subscript "s" refers 

to ions in the surface layer ; 

(2) (a) H.o + OH/— ->HOH, 

(6) H/>+ OH, 1 — *HOH— the subscript «f» refers 
to the freely moving ions in the liquid. 
In contact with pure water, neutralizations according 
to scheme 2 are small in number. Also 2 (a) and 2 (b) are 
equally probable. Consequently the numbers of H s ° and 
OH, 1 remaining in the surface at any instant will be equal, 
and the surface will be neutral. 



the Adsorption of Ions. 333 

When an acid is added to the water the neutralizations 
according to scheme 2 (a) will he completely negligible, but 
those according to scheme 2(b) will not be so. The total 
number of neutral molecules of water formed in the surface 
is still equal to " .r," but a number of them is now being 
formed according to 2 (/>). Corresponding to the number of 
neutralizations according to 2 (6), a number of hydrogen 
ions will remain in the surface layer in excess of the number 
of hydroxyl ions. The rate at which 2 (/>) proceeds thus 
determines the free charge on the surface. An equivalent 
number of anions remain unneutralized in the liquid and 
form the second mobile sheet of the double layer. 

The free charge on the surface will evidently increase with 
rise in the concentration of hydrogen ions in the solution. 
There are, however, two factors opposing this increase in the 
charge of the surface. 

A. The proportion of hydrogen ions striking on the surface 
diminishes as the positive charge of the surface increases. 
Only those ions which have sufficient kinetic energy to over- 
come the electrical repulsion can reach it. If e be the 
potential of the double layer in C.Gr.S. units, then the number 
of collisions of the ions per unit surface per second is pro- 
portional to 

U H o.C E o.^- E /^ ..... (6) 

where H o denotes the concentration of free hydrogen 
ions in the liquid, "E" is the electronic charge in C.Gr.S. 
units, T is the absolute temperature, Uho is the mobility of 
the hydrogen ions in water, and K = E/No, where " R " is the 
gas constant and No the Avogadro number. 

B. The other factor that tends to diminish the charge of 
the surface is the electrical adsorption of the anion of the 
acid added to the solution. That this plays an important 
part will be evident from the following examples taken from 
the observations of Perrin : — 

Rate of Electro- 
Substance. Electrolyte. endosraotic outflow. 

XUOr, M/1000 HC1 +110 

„ M/1000 citric acid + 5 

,, M/1000 HNO :i (or HOI) +100 

„ M/1000 H 2 S0 4 + IS 

OCl 3 M/1000 IIN0 3 + 85 

„ M/1000 H.^SO, + 21 

,. M/500HO1 + 90 

, M/1000 H, C 2 , + 30 

„ Feebly acid with HC1 + 75 

., Solution of KH, (P0 4 )1 

with approximately the 
same number of free 
hydrogen ions as above 



334 Prof. J. N. Mukherjee on 

Both these factors tend to diminish the rate of increase of 
the charge with rise in the concentration of hydrogen ions. 
For acids with simple univalent anions, the electrical ad- 
sorption at low concentrations can be left out of account in 
view of the excessive mobilty of the hydrogen ions. 

A quantitative relationship can now be obtained between 
the charge on the surface and the concentration of the acid. 
Let x 1 be the rate of neutralization according to 2 (6) above. 
We have then 

^K.^.Cho.-*- 3 ^, .... (7) 

where k is a constant. 

The density of the charge on the surface is proportional to 
x 1 — which is a measure of the number of hydrogen ions 
remaining in excess in the surface. If the thickness of the 
double layer remains constant then the potential of the 
double layer is proportional to the density of the charge : 
that is, to x 1 . 

When all the hydroxy 1 ions in the surface layer are being- 
neutralized according to 2 (b) the surface will have a maximum 
charge determined by i( x." 

Putting x\/% = 0, since u x" is a constant, we have 

e*6, (8) 

and 6 represents the ratio of the hydrogen ions present in 
excess at any instant in the surface layer to the maximum 
number possible when the neutralization takes place only 
according to 2 (b). The potential of the double layer can be 
written as 

e = h . x 1 = k 2 . 6 = h . Oho . e-*- ^ . U H o, . . (9) 

or # = £ .Cho.^-*/ t .U H o, (10) 

where k\, Ic 2 , fe, Ic and /3 denote constants. 
Similarly, for alkali solutions we have 

e=ko.Gom.e-^^.UoKi. ...... (11) 

The maximum charge, being determined by a, will be the 
same with alkali as with acid. Of course, the influence of 
the oppositely charged ion in the acid or the base is being- 
neglected. 

Case 2. — 2 he surface is not chemically inert : preferential 
adsorption of one ion is possible. 

A review of the literature shows that surfaces in contact 
with water are seldom neutral. They are generally more err 
less negatively charged. This is intelligible in view of the 






the Adsorption of Ions. 335 

chemical reactivity of the hydroxyl group. The presence of 
the potentially tetravalent oxygen atom possibly leads to a 
selective adsorption of hydroxyl ions by most surfaces. Thus 
glass and quartz have a marked negative charge in contact 
with water (cp. Elissafoff). On the addition of an acid the 
electrostatic forces will produce a diminution of the charge. 
The electrical adsorption of hydrogen ions by hydroxyl ions 
cannot be distinguished from the recombination of hydrogen 
and hydroxyl ions to form neutral molecules of water. This 
is confirmed bv the fact that the equation of electrical ad- 
sorption (cp, previous paper) satisfactorily represents the 
diminution of the charge. 

Perrin [loc.cit) found that, excepting alumina and chromium 
chloride, all other substances (naphthalene, silver chloride, 
boric acid, sulphur, salol, carborundum, gelatine, and cellulose) 
show a preferential adsorption of hydroxyl ions. The sur- 
faces have a negative charge even in contact with acid 
solutions. He also found that at higher concentrations of 
the acid the surface acquired a positive charge. Elissafoff, 
McTaggart, Ellis, Powis, and others could not observe this 
reversal in their investigations. Electrical adsorption of 
hvdrooen ions cannot lead to a reversal of the charge. The 
reversal (or the non-reversal) of the charge becomes intelli- 
gible if it is assumed that the considerations set forth in 
deducing equations (8) or (9) are correct. 

In contact with pure water the surface has a layer of 
adsorbed water and a number of hydroxyl ions. The amount 
of hydroxyl ions adsorbed by the surface will, in general, be 
small, as the concentration of the hydroxyl ions is very 
small in pure water. If, however, the adsorption is very 
strong the surface will have a considerable negative charge. 
On the addition of an alkali the negative charge of the 
surface will increase, due to two reasons: 

(1) the preferential adsorption of hydroxyl ions will 
increase, and 

(2) the number of hydrogen ions being formed at the 
surface will be more and more neutralized by hydroxyl ions 
in the liquid (cp, scheme 2 {a)). A maximum will be 
reached when the surface is saturated by preferential ad- 
sorption and when — \ in equation (9). The maximum 
charge per unit area can be written as 

E ro =# + y (for alkali), .... (12) 

where " os" corresponds to the charge when = 1 in 
equation (8) and " y" is proportional to the number of 



336 Prof. J. N. Mukherjee on 

hydroxyl ions the surface can adsorb per unit area when it is 
saturated. 

Since the chemical adsorption of hydrogen ions is assumed to- 
be absent, on the addition of an acid the negative charge will 
decrease owing to electrical adsorption till the surface becomes 
neutral. At this concentration of the acid, the surface has 
an adsorbed layer of water, and an equal number of hydrogen 
and hydroxyl ions, An increase in the positive charge 
cannot be due to electrical adsorption of the univalent 
hydrogen ions (cp. previous paper). The increase in the 
charge is due to the neutralization of the hydroxyl ions 
being formed in the surface by impinging hydrogen ions, as 
represented in scheme 2 (/>) above. 

The maximum charge E 9rt for an acid will, therefore, be 
equal to " x, 9i 2 he maximum charge due to acids thus gives 
a measure of the hydration of the surface. The difference 
between the maximum charge observed with acid and with 
alkali gives a measure of the amount of hydroxyl ions that 
is required to saturate the surface. 

In trie preceding discussion, the chemical and electrical 
adsorption of the anion of the acid lias been left out of 
account for the sake of simplicity. If the initial negative 
charge of the surface in contact with pure water is consider- 
able the electrical adsorption can be complete only at high 
concentrations of the acid, i. <?., the surface will be neutral at 
a high concentration of the acid. The electrical adsorption 
of the anion, is no longer negligible. A reversal of the charge, 
though theoretically possible, may not be actually observed 
owing to the great concentration of the anion. 

The reversal is thus dependent on : — 

(1) a. large value of x, and 

(2) a small value of y. 

A non-reversal is to be expected when the opposite is the 
case, ?'. <?., 

(1) a small value of " x" and 

(2) a large value of "?/•" 

A regular transition from marked reversal to non-reversal 
can be observed in Ferrin's work. With cellulose he also 
does not record a reversal of the charge. It is to be expected 
from the preceding considerations that non-reversal will not 
be observed when the concentration of the acid required to 
render the surface neutral is comparatively high, i. <?., the 
anion concentration is high. The concentration of the acid 
in the case of cellulose is the greatest recorded by Perrin. 



the Adsorption of Ions. 337 

The chemical adsorption of the anion is also not to be 
neglected. The experimental data on this subject are meagre. 
The various points raised here can be experimentally eluci- 
dated. As shown above, the standpoint developed in this 
paper can correlate all the observed tacts. Besides, it gives- 
a definite idea of the electrical double layer. 

Adsorption of electrolytes. 

In the preceding sections the adsorption of ions has been 
considered with reference to the electrical charge of surfaces- 
in contact with aqueous solutions of a single electrolyte. 
The electric effects accompanying the adsorption of ions have 
enabled us to follow the total adsorption of ions of both signs. 
In considering the adsorption of ions measured by chemical 
means it is important to remember the influence of the ad- 
sorption of the solvent pointed out by Arrhenius, Bancroft, 
Williams, and others. 

The amount adsorbed is small and the analytical measure- 
ment is difficult. For this reason, investigations have centred 
round adsorbents with great adsorbing power and substances 
which are strongly adsorbed. Often it happens, that if a sub- 
stance is used in a satisfactorily pure state it does not have 
the necessary specific surface to make the estimation of the 
adsorbed amount possible. As a result adsorbents generally 
contain small amounts of other substances. The importance- 
of these impurities has been pointed out by some investigators. 

Michaelis and Freundlich and their co-workers have done 
systematic work in this field. Their investigations have 
brought out the following regularities : — 

(«) The electric charge of the solid influences the ad- 
sorption, Thus Michaelis and Lachs (Z. Elektro-Cliem. xviL 
pp. 1, 917 (1911)) ; Biochem. Zeitsch. xxv. p. 359 (1910) ) x 
and Davidsohn, Biochem. Zeitsch. liv. p. 323 (1913)) found 
that in contact with acid solutions chnrcoal adsorbs anions 
strongly and does not adsorb cations. The reverse happens 
in the case of cations. Freundlich and Poser [Koll. Cliem.. 
Beihefte, vi. p. 297 (1914)) undertook an extensive investiga- 
tion, and they agree with Michaelis as to the electro-chemical 
nature of the adsorption. 

(l>) The chemical nature of the adsorbent has a specific 
action. 

Michaelis and Rona {Biochem. Zeitsch. xcvii. pp. 57, 85 
(1919)) believe that adsorption is due to chemical affinity. 
They mention that charcoal has a great capacity for adsorbing 
substances containing a chain of carbon atoms. (Cp. Abder- 
halden and Fodor, Fermentforschuncj, ii. p. 74 (1917).) 
Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. ^ Z 



338 Prof. J. N. Mukherjee on 

Freundlich and Poser (loc. cit.) found that the nature of the 
adsorbent plays an important part in determining the ad- 
sorbability of a dye. 

Both Michaelis and Freundlich agree that at least two 
types of adsorption of ions can be recognized. 

(c). Exchange or displacement of ions already adsorbed by 
ions of a second electrolyte (cp. Freundlich, ie Verdrangende 
Ionenadsorption " and Michaelis, "Austausch-Adsorption "). 
Michaelis (Z. Electrocliem. xiv. p. 353 (1918)) considers that 
a substance like mastic, or kaolin (bolus), acts as a " zweier 
electrode " (a binary electrode). Thus kaolin has a slow- 
moving anion (silicate ion) anchored on its surface and tends 
to send hydrogen ions into the solution under a definite 
electrolytic solution tension. Freundlich points out (and 
Poser, loc. cit.) that other cations can displace the hydrogen 
ions and form undissociated complexes (and Elissafoff, Z. 
Phys. Chem. lxxix. p.. 385 1912). 

(d) An adsorbent which contains some adsorbed electro- 
lytes need not be necessarily saturated. In this case, besides 
an exchange of ions, primary adsorption of ions is possible. 
This also applies to substances which act as binary electrodes 
in the sense in which the word has been used by Michaelis. 
He considers that, besides adsorption through exchange of 
ions, there is only one other type of adsorption, namely, 
adsorption of both ions in equivalent amounts (" Aquivalent 
Adsorption"). 

One other fact has been emphasized by these authors. 
(e) It is the irreversible nature of electro-chemical ad- 
sorption. The well known instance of the adsorption of 
hydrogen sulphide by metal sulphides studied by Linder and 
Picton (T. Ixvii. p. 163 (1895) ; Whitney and Ober, J. Amer. 
Chem. Soc. xxiii. p. 842 (1901)) can be mentioned. The 
adsorbed substance does not come out in solution when the 
adsorbent is brought in contact with pure water. 

(/) Lastly, there is no clearly established instance in 
which hydrolytic splitting up of neutral salts such as 
potassium chloride has been observed through adsorption. 

Theories regarding tlie Exchange of Ions. 

The conception of an adsorbent acting as a binary electrode, 
suggested by Michaelis, is not of much help in explaining the 
exchange of ions and other peculiarities of the adsorption of 
electrolytes. The relationship between the adsorption of 
ions, electro-endosmotic cataphoresis, and precipitation of 
colloids has been established beyond doubt. The only theory 



the Adsorption of Ions. 339 

that attempts to correlate them is that due to Freundlich. 
This view is an extension of Michaelis's idea referred to above. 
The adsorbent (or colloidal particle) is regarded as a great 
multivalent ion (dp. Billiter, Z. Phys. Chem. xlv. p. 307 
(1003); Duclaux, J. Chim. Phys. v. p. 2\) (1907)). The 
following extract shows clearly their standpoint (Freundlich 
and Elissafoff, lor. cit. p. 411) : — 

"Die Ladung soil nun durch die verschieden grosse 
Losungs tension der lonen des schwerloslichen festen Stoft's 
zustande kommen, aus dem das suspendierte Teilchen, bzw. 
die Wand besteht. Nimmt man als Beispiel das Glas, so hat 
man an der Oberflache desselben eine Schicht von gelostem, 
oder bzw. in wasser gequollenem Silkat ; die K- and Na- 
lonen haben eine orosse Losunostension und bilden eine 
aussere Schicht, die schwerloslichen, langsam diffundierenden 
(vielleicht auch stark absorbierbaren) Silikationen bilden 
eine innere Schicht, die mit dem festen StotF verbunden wie 
ein vielwertiges Ion sich verhalt. Der wesentliche Unter- 
schied o-egen ein oewohnliches Ion lieut darin, dass wegen 
der Grosse GrenzfTachenwirkungen eintreten, die Ronzentra- 
tion ist in der Umoebuno- nicht so homooen, sondern es sind 
durch Adsorption hervorgerufene Konzentrationsunterschiede 
vorhanden. 

"Fiir zwei ionen gilt nach der Massenwirkungs gesetzt 
(Anion) . (Ration) = k (nichtdissociertes Salz), deshalb auch 
fiir das mikronische, vielwertige Anion des als Beispiel 
betrachteten Glases. 

" (Vielwertiges Anion) . (Ration) = R (nichtdissocierter 
Stoff). Es wird also von der Ronzentration der Rationen die 
' Ronzentration des vielwertigen Anions/ d.h. auch die Zahl 
der auf der Grenzflache vorhandenen Ladungen abhangen. 

"Die Rationenkonzentration, am die es sich hier handelt, 
wird aber in erster Linie die der nachsten Umoebuno- der 
Grenzflache, d.h. die Adsorptionschicht sein. Die adsorbierte 
Menge Ration wird also fiir die ' Ronzentration des viel- 
wertigen Anions' d.h. fiir die Laduno- der Grensflache niass- 

. ... 

gebend sein. Dies ist eine andere Verkniipfung von 

Adsorption und Potentialdifferenz an der Grenzflache. 
Genau das Gleiche gilt naturlich fiir ein vielwertioes Ration 
und die adsorbierten Anionen." 

There are several difficulties in accepting this theory. 
Salts of alkali metals can neutralize charged surfaces at 
moderate concentrations (N/10 or N/20). One has to con- 
clude that the alkali salts of these " multivalent anions " have 
a low solubility product. The effect of the valency of the 

Z2 



340 Prof. J. N. Mukherjee on 

oppositely charged ion cannot be accounted for. The activity 
of the cations is generally in the following order : — 

Th > Al > Ba > Sr > Ca > H > Cs > Rb > K > Na > Li. 

The postulates that alkali metal salts become undissociated 
at low concentration of the cation and that their solubility 
products are of the above order for a large number of diverse 
chemical substances, are contrary to experience. Regarded 
from the chemical point of view the generality of these ob- 
servations cannot be explained. Besides, the conception of 
the suppression of the dissociation of a salt cannot explain the 
reversal of the charge which is met with when the oppositely 
charged ion is polyvalent. 

The view of electrical adsorption put forth by the writer 
gives a definite correlated account of these various facts. 

The Role of Electrostatic Forces in the Absorption of Ions. 

(a) Adsorbent in contact with a single electrolyte : — 
Let us consider an adsorbent, P, in contact with an electro- 
lyte A B . It is assumed that the substance P only adsorbs 
the anion B by chemical affinity. For simplicity it is also 
assumed that " P " is a pure chemical substance of definite 
composition. The amount of B adsorbed per unit area will 
depend on the concentration of A. B and on the strength 
of the chemical affinities acting on B _ . Corresponding to 
the number of anions adsorbed an equivalent number of 
cations A remain in the solution. These are held near the 
surface by electrostatic forces, and form the second mobile 
sheet of the double layer (cp. the earlier paper referred to). 
If the concentration of the electrolyte is sufficient, some of 
them will be fixed on the surface by electrostatic forces. 
These ions of opposite charge fixed on the surface by electro- 
static forces will be spoken of as electrically adsorbed in 
the sequel. The chemical adsorption of an ion thus concen- 
trates both ions at the surface in equal amounts. That is, 
the primary adsorption is an equivalent adsorption of both 
ions. Analytical methods cannot differentiate between the 
two adsorptions, but electro-osmotic and cataphoretic experi- 
ments can (cp. (d) above). 

If the adsorption of the anion is due to strong chemical 
forces, perceptible amounts of the electrolyte A B ~ will be 
adsorbed at very low concentrations. Even saturation may 
be reached at low concentrations. In such cases, if the 






the Adsorption of Ions. 341 

adsorbent with adsorbed electrolyte is suspended in pure 

water the adsorbed electrolyte will not be set free (e). 

Since the primary adsorption of the ions is due to chemical 

affinity, the influences of the nature of the adsorbent and of 

the electrolyte (b) are intelligible. 

(/>) The addition of a second electrolyte : — 

The general case when both electrolytes, A B~ and 

C D , are present in all possible concentrations will be too 

complex. It will be assumed for the sake of simplicity 

that— 

(1) the substance P adsorbs chemically the anion B^ 
strongly, and that the concentration of the electrolyte A B 
in the liquid is negligible. We are thus dealing with an 
adsorbent with an amount of adsorbed electrolyte in contact 
with a second electrolyte solution ; 

(2) tiie atoms on the surface of the adsorbent P do not 
exert any chemical affinity on the ions C and D~. 

This particular case corresponds with most actual systems, 
and the electrolyte A + B~ plays the part of the " Aktiver 
Electrolyt " of Michaelis. 

Let us now consider the effects of the electrostatic forces 
on the ions C and D~. A cation C , when it- diffuses into 
the double layer owing to thermal energy, will be attracted 
to the surface. Considering the kinetic equilibrium between 
the ions in the second sheet of the double layer (A and C ) 
and those in the liquid, it is evident that the relative propor- 
tion of A and ions in the double layer will depend on 
(i.) their respective concentrations in the bulk of the liquid, 
and (ii.) their valency. The same consideration applies to 
the electrically-adsorbed ions A + or C + . At sufficiently 
large concentrations the whole of the mobile second layer 
and electrically-adsorbed ions will be formed by the ions 
C . There will thus be an exchange of ions, and the amount 
of exchange will depend on the concentration of the second 
electrolyte. When the displacement is complete the amounts 
exchanged will be equivalent to the amonnt of B* ions 
primarily adsorbed and independent of the nature of the 
replacing ion C + — a fact often observed (cp. iLinder and 
Picton, loc. cit. ; Whitney and Ober, loc. cit. etc.). 

The ions will be positively adsorbed. 

The relationship between the charge of the surface and 
the positive adsorption of the oppositely-charged ion is also 
obvious. The amount of () + ions absorbed depends on the 



342 Prof. J. N. Mukherjee on 

amount of the negative ions chemically adsorbed. Anything 
that increases the total amount of adsorbed negative ions 
will increase the positive adsorption of C • 

The reverse case, when the positively-charged ions A + are 
adsorbed chemically instead of the ions B~? and no other 
ions are chemically acted on by the surface, is obvious. 
Negative ions will now be positively adsorbed and exchanged. 
This state of affairs corresponds with the statements made in 
(a) and (c) above. 

Taking the same case again, we shall consider the effect 
of the electrostatic forces on the ion D • An ion D~ , diffusing 
into the double laver, will be driven out of it. So long ;is 
the potential of the double layer is sufficiently strong, a 
volume of the liquid equal to "SI," where " S " is the extent 
of the surface and " I " is the thickness of the double layer — 
will be free from the ion D . In other words, the concen- 
tration of D~ increases in the bulk of the liquid and a 
negative adsorption will take - place. This will increase with 
the concentration of the electrolyte so long as the potential 
of the double layer is sufficiently strong. Since with in- 
crease in concentration the potential fall^, the negative 
adsorption will reach a maximum. At concentrations when 
the surface becomes electrically neutral, there should be no 
negative adsorption due to electric forces. It is difficult to 
determine negative adsorption at high concentrations as the 
osmotic pressure opposes it. Also, the variations in concen- 
tration due to negative adsorption become relatively small. 
The experimental difficulties lie in the analytical estimation 
of small amounts. Only ions which can be estimated in 
extremely small amounts are suitable for experiment. 

Estrup (Koll. Zeitscli. xi. p. 8 (1911)) has actually 
observed a negative adsorption of the oppositely-charged 
ion. He estimated the adsorption of the iodate, dichromate, 
and chromate of ammonium. Michaelis and Lachs (KolL 
Zeitscli. ix. p. 275 (1911)) did not observe a negative 
adsorption with potassium chloride. 

Exactly similar observations have recently been made by 
Bethe (Wiener Mediz. Wochsch. 1916, Nr. 14 ; Koll. Zeitsch. 
xxi. p. 47 (1917)). He worked with gelatine gel, gelatine 
sol, and a number of animal cells. The adsorption of a basic 
dye is greater in weak alkaline solutions than in neutral 
solutions. The same is the case for an acidic dye in weak 
acid solutions. In alkaline solutions the adsorption of acid 
dyes is negative, and the same is the case with basic dyes 
in acid solutions. Examples of the role of- the electrical 



the Adsorption of Ions, 343 

force in the adsorption of ions can be multiplied (cp, Baur, 
Z.Phijs. Chem. xcii. p. 81 (191(5) ; Michaelis and Davidsohn, 
toe. cit.). 

Exchange of Bases in Soil and Soil- Acidity 

It is now easy to understand the nature of the exchange of 
bases in soil-analysis and the cause of soil-acidity. A com- 
plete reference to the older literature is given in the following- 
papers : — 

(1) McCall, Hildebrandt, and Johnson, J. Phys. Chem. 
191b", xx. p. 51. 

(2) Rxce; ibid. p. 214; (3) Truog, ibid. p. 457. 

Russell (Brit. Assoc. Rep. 1918, p. 70) has given an 
excellent summary of the present position of the subject. 
The facts are that — 

(a) Neutral solutions of salts like potassium chloride, if 
treated with samples of soil, give acid extracts though the 
extract with pure water is neutral. 

(//) In a large number of cases it has been shown that 
there is a definite exchange of the cations. Equivalent 
amounts of bases are exchanged in many cases. 

Two different views have been advanced to explain the 
facts, The older chemical view regards the process as a 
chemical interaction between definite acids (e. g., humus 
acid) or complex salts {e.g., silicates) and salt solutions. 
The other view begins with von Bemmelen, and regards it 
as an adsorption process. Cameron suggested (cp. Russell's 
Report) that the soil adsorbs the base more strongly than it 
adsorbs the acid. 

The objections against the chemical view can be sum- 
marized as follows : — The extract wich pure water being 
neutral, the soil-acids must be insoluble. The acids must be 
unusually strong, as they evidently decompose a neutral 
salt solution combining with the base, liberating the strongest 
known acids, like hydrochloric acid. 

Evidently such acids are unknown, and it is difficult to 
conceive of such reactions. Regarding the exchange of 
buses, the difficulty lies in the assumption that the basic ion 
is taken up to form an insoluble salt. It is necessary to 
postulate the existence of insoluble salts of alkali metals in a 
large number ot' cases (cp. the remarks on Freundlich's 
theory). 

That adsorption plays an important part is also evident 
from the works of: Russell and Prescott (J. Agric. Sci. viii. 
p. 05 (191G)) on the interaction of dilute acids and phos- 
phates present in the soil. But the view of: Cameron does 



344 Prof, J. N. Mukherjee on 

not seem to be tenable. The preferential adsorption of an 
ion by the soil does not mean hydrolytic decomposition of 
the salt. It appears from the summary given by Russell 
•that the equivalent exchange of bases lies in the way of 
regarding the reaction as an adsorption process (loc. cit. 
pp. 71, 75 ? 76). It would be apparent from the previous 
discussion that this, in itself, does not contradict the adsorp- 
tion hypothesis. 

Soil can be regarded as a complex colloidal system. It is 
a complex gel consisting of aluminium and other silicates, 
free silica, ferric hydroxide, etc. The gel is mixed with 
insoluble crystalloids. It also contains small quantities of 
adsorbed electrolytes and organic matter in indefinite and 
varying proportions. The gel adsorbs anions by chemical 
affinity. These anions may be : — 

(1) of organic acids, such as humus acid ; 

(2) of simple electrolytes like chlorides, sulphates, car- 

bonates, etc. ; 
(3) hydroxyl ions from water. 
. Owing to the complex chemical nature of the gel and the 
enormous specific surface of gels, large quantities of anions 
may be adsorbed. An equivalent number of cations remain 
near the surface as the mobile second sheet or as electricalty 
adsorbed. The exchange of bases is simply due to the dis- 
placement of these ions. When the displacement is quantitative 
equivalent amounts are exchanged. The anions primarily 
adsorbed or the cations in the second sheet are not of one 
kind. The relative numbers and chemical natures of these 
ions will evidently vary with the different soils. 

An extract with pure water will be neutral unless the soil 
contains free acids. An extract with a neutral salt can only 
be acid when the cations displaced from the second sheet (or 
electrically adsorbed) contain hydrogen ions or such ions as 
aluminium, which hydrolyse in dilute aqueous solutions. 
The role of the aluminium ions in determining the acidity of 
the soil extract has been pointed out by Daikuhara (Bull. 
Imp. Central Agri. Expt. Station, Tokio, ii. pp. 1-40 (1914)), 
and has been fully confirmed by Rice (loc. cit.). The function 
of organic acids lias constituted a great objection against the 
adsorption hypothesis. The hydrogen ions in the second 
sheet have probably, in most cases, their origin in these 
acids. This view thus correlates the exchange of bases 
observed with soil with such exchanges as have been observed 
in the adsorption of electrolytes (cp. Michaelis). 

That sometimes considerable quantities of bases are ex- 
changed should be referred to the enormous surface of these 






the Adsorption of Ions. 345 

gels, and that probably the surface is saturated with anions. 
As crystalloids (insoluble) are also present, the type of 
exchange considered by Paneth (loc. cit.) is also possible. 

It is needless to point out that in this discussion only the 
theoretically simple case has been considered. Complications 
due to simultaneous primary adsorption of different ions and 
their mutual displacement are not always negligible. Besides, 
the changes may not be restricted to the surface ; formation 
of solid solutions, etc., are not excluded. Considering all 
these complex influences, it is interesting to note that most 
of the observed regularities correspond to the theoretically 
simple case. 

Adsorption of Ions in its Relation to Permeability of 
Membranes and to Negative Osmosis. 

In conclusion, a few remarks will be made on the funda- 
mental interest that a study of the adsorption of ions has for 
biological phenomena. Cell activity is greatly conditioned 
by the permeability of its "walls" or the cell-substance to 
the contents of the liquid with which it is in contact. The 
connexion between the rate of osmotic flow through mem- 
branes and even the direction of the flow, and the potential 
differences existing on the two sides of the membranes, has 
been clearly established (Girard, C.R. cxlvi. p. 927 (1908), 
and following authors : Bartell, J. Amer. Chem. Soc. 
xxxvi. p. 6±6 (19.14) ; Hamburger, Z. Pliys. Chem. xcii. 
p. 385 (1917)). The origin of the potential difference 
is generally assumed to be due to the fact that the rate of 
diffusion of the electrolytic ions in the membrane substances 
is different from that in water. That the membrane potential 
is due to a selective permeability of ions was first suggested 
by Ostwald (Z. Phys. Chem. vi. p. 71 (1890)) ; Donnan (Z. 
Elektrochem. xvii. p. 572 (1911)) has discussed the origin 
of the potential differences theoretically, and has given it a 
quantitative form based on thermodynamic considerations. 
In collaboration with others he has carried out a number of 
investigations which have established the validity of this 
view. 

The simpler case of a potential difference between two 
interfaces when an immiscible liquid is placed between 
two aqueous solutions has also attracted a good deal of 
attention. The work of Loeb and bis co-workers on cell- 
permeability and origin of the membrane potential is of 
fundamental importance (Loch and Beutner, Biochetn. Zeit. 
li. p. 295 (1913) ; Beutner, Z. Phys. Chem. lxxxvii. p. 385 



346 Prof. W. M. Hicks on certain Assumptions in the 

(1914), Z. Elektrochem. xix. pp. 329, 473 (1913) ; Loeb, 
J. Gen. Phys. xx. p. 173 (1919), ii. pp. 273, 255, 387, 577, 
563, 673, 659). The part played by the adsorption of ions 
in these phenomena is twofold. The origin of the potential 
is in many instances due to the adsorption of ions (cp. Baur, 
Z. Elektrochem. xix. p. 590 (1913) ; Z. Phys. Chem. xcii. 
1916, p. 81). 

Secondly, the electrostatic forces of the surface probably 
determine the relative permeabilities of the two ions. To 
this the semi- permeability of an ion can be referred. 

Regarding negative osmosis, attention may be drawn 
to the suggestion of Freundlich [Koll. Zeitsch. xviii. p. 1 
(1916)) that the thin walls of the membrane substance 
conduct electricity, and electro-osmotic flow of the liquid 
occurs. A necessary condition is that one ion is permeable 
and the other relatively impermeable. This explanation 
meets thermoclynamical requirements, and is the only satis- 
factory one hitherto put forward. 

In all these cases the same influences of polyvalent ions 
and ions of opposite charge are noticeable. 

The change in the collodial properties of the membrane is 
an important additional factor which has to be remembered. 
The influence of the electrostatic forces is unmistakable. 

Physical Chemistry Department, 

University College, London. 



XXX. On certain Assumptions in the Quantum-Orbit Theory 
' of Spectra. By W. M. Hicks, F.R.S* 

THE practically complete success of the quantum-orbit 
theory in describing all the known facts of spectra, 
in cases where we know experimentally that the source 
consists of a single nucleus and a single electron, must 
give assurance that the same procedure must also be capable 
of application to more complicated atoms than those of the 
hydrogen and enhanced helium types. Unfortunately, 
however, mathematical difficulties have so far prevented 
any rigorous application of the theory to definite cases, 
even of the next simplest atomic configuration of a single 
nucleus and two electrons. The attempt of Sommerfeld 
at an approximate solution shows, on the one hand, how 
hopeful we may be of a description of spectra on this basis, 
and at the same time how far we are at present from its 
* Communicated by the Author. 



Quantum^ Orbit Tlieory of Spectra. 347 

achievement. In the present note I wish to illustrate this 
by drawing attention to certain assumptions as to actual 
spectral data, which have been made and which do not 
appear to be justified. The criticisms may not affect 
essential points, but they would appear to require some 
modification in the presentment of the tlieory. References 
will be made to Sommerfeld' , s 'Atombau und Spektrallinien,' 
2nd edition (1921). 

1. Sommerfeld (pp. 27G, 50b') takes a configuration of 
a central nucleus, surrounded by a ring of equally-spaced 
electrons, and at a considerable distance furl her out one 
electron revolving in a quantized orbit. On fhe assumption 
— here justified — that the ring can be treated as if the 
w r hole charge of the electrons on it were continuously 
distributed along it, he obtains as an approximation the 
same form for a sequence function (or term) p as that' 
suggested by Ritz, viz.* : 

He says that this is the actual true form, as already deter- 
mined by observation. This is, however, by no means the 
case. No form has yet been found which will fit in for all 
series, and indeed the form N/(m + /u. + a/m) 2 is in general 
rather superior to that of Ritz. It is to be noted that the 
assumption made above leads to the same result as if the 
force to the centre depended only on forces inversely as 
even powers of the distance, and forces depending on odd 
powers — say l/r' d — are excluded. It may also be noted in 
passing that the theory so developed applies only to the 
case of a single external electron and one internal ring, 
that is, on the usually assumed configuration of eight- 
electron rings, only to the spectra of the fluorine group, 
or the ionized rare gases, or the doubly-ionized alkalies, etc. 
By taking his E as (k—sj^e in place of ke, the formula 
would meet the more general case. This modification, 
however, would only slightly affect the order of magnitude 
of the quantities /a, a. 

In the formula m=n-\-n', where n, n are respectively 
azimuthal and radial quantum numbers, and //,, a arc 
functions of n and not of n'. 

* As a result of successive approximation, « being small, -this means 
for a complete approximation the form 

which, as is well known, is capable of reproducing- practically all cases 
if /*, a, {3 . . . are all at disposal, and are not related necessarily to 
one another, as here. 



348 Prof. W. M. Hicks on certain Assumptions in the 

It is not to be expected that the numerical values of the 
constants fju, a. on this special theory should accord with any 
determined by experiment, but they should be of a suitable 
order of magnitude and general character. It may be 
interesting to test this. The expressions for the constants- 
//,, a may be written 

8 ( ,15/3/ 3*-*A ,1 2 



where 

, _ (27r) 4 m 

4/< 
and 

<x 3?r 



^ = (2,)%Ve ( z^ ) = 8 . 9(Z _^ )( ^^ )ioiv 



/9_Z-*x 2 
/* " 2n 2 h\2 k-sic) 7 ' 

If p be measured in wave number instead of frequency, 
the a must be multiplied by the velocity of light. Then 

uc 6-12 /9 Z-£ 



/9 &-k \ 
\2 k-sj 



/jl n" \4 /c — Sk 

Here r denotes the radius of the internal ring in cm., Z the 
atomic number of the element, k the number of external 
electrons, and s k depends on the mutual action of the external 
electrons on one of them. For Li, Z = ll, & = 3, s/ >; = , 577, 
and 



2-2/. 4-5 



10 



16, 



In actual cases, for wave numbers of about p = 10 5 , poLc/ji 
lies between about *9 and *01. Hence the second equation 
requires r< 10~ 7 ' 5 > 10 -8 ' 5 . Since fi<l, the first equation 
requires r to be about 10 -8 , but as the second term in the 
bracket is determined by an approximation it must be a 
small fraction, whence r < 10 -8 ' 5 . The fact that both give 
values of the same order of magnitude, even if they cannot 
exactly agree, and not far from what might be expected for 
an 8-electron ring, is certainly satisfactory. 

2. It is deduced that the different types of sequences 
correspond to azimuthal numbers n — 1, 2, 3, 4, the different 

orders of the same type to radial quanta n' = 0, 1, These 

are then co-ordinated with the s, p, d, f types because it is 
stated that these types have their lowest orders respectively 
of 1, 2, 3, 4. It is difficult to see how this statement has 
been arrived at, as it is quite incorrect. For the sake of 



Quantum- Orbit Theory of Spectra. 34',) 

readers who may not be familiar with spectral data, it may 
be well to consider them here. 

(s, p.) For the s, p the lowest order are : 

s. p. 

TT [ Alkaline earths ... 2 1 

"'iZn, Cd, Eu, Hg... 2 1 



p. 

Rare gases 1 1 



T [The alkalies I 2 

L - \Cu, Ag, Au 1 J 



Group III 2 1 



I£ it were not for the cases of the rare gases and the 
Cu subgroup, the assumption might be explained by an 
interchange of the nature of the sequences which produce 
P, S series (for which in Groups I. and IJ. indeed there is 
also direct evidence). But that two groups make s, p both 
have unity for their first order is fatal. 

(r/.) The assumption of 3 as the first order for d(in) 
no doubt is based on the fact that Pitz made it in dealing 
with the D series in the alkalies. The denominators of the 
first orders in this group are comparable with 2*9, which 
Ritz wrote as 3 — '1 and called the first order 3. But this 
procedure is inadmissible either on the side of the formula or 
from what Ave know of the constitution of the d sequence. 
In Sommerfeld's formula /jl is positive, and it is only by 
treating the fraction as positive that we find a detinite 
dependence of it on certain spectral constants. But even so, 
the first order is not 2 for all groups. The law of the first 
order of the d sequence is a quite simple and definite one, 
and is given on p. 18S of my recently published 'Analysis of 
Spectra.' It is that in each group of the periodic series, 
the subgroup of elements whose melting-points increase with 
atomic weight take as their first order m = l, whilst the sub- 
group with decreasing melting-points take m = 2. 

(/.) In the case of / the 3 + fraction has again clearly 
been written 4 — /, and the assumption lias been made that 
the lowest orders of the / type take m = 4. But here also, 
for the same reasons as in d, the fraction must be taken as 
positive. In the alkalies certainly the lowest order observed 
is F(3), but F(2) would lie far up in the ultra-red, beyond 
even Paschen's longest lines. In the Cu subgroup there is 
evidence for m = 2 and indications for m = l. The alkaline 
earths have m — 2 both in triplets and doublets. In the 
Zn subgroup only F(3) has been observed, but F('2) would 
lie in the extreme ultra-red. In Group III. there is no 
evidence, whilst in the rare gases there are examples of* F(l) 
and F(2). 

It would thus appear that the theoretical deduction that 
different types depend on successive changes of azimuthal 
quanta by unity is not tenable. 



350 Prof. W. M. Hicks on certain Assumptions in the 

3. In dealing with the Zeeman effect on p. 422, Sornmer- 
£eld adopts Paschen and Back's interpretation of their 
experiments on the Zeeman effect in the case o£ close 
multiple lines. This interpretation was based on precon- 
ceptions as to the nature of the series types in He and 
Li, which they investigated. I have given * reasons why 
this interpretation should be modified. On either inter- 
pretation, however, a consequence follows which appears 
difficult to explain on the quantum-orbit theory. Take, for 
example, the case of the helium doublet at 4713 A. Each 
component in weak fields shows special Zeeman patterns. 
W4th increasing fields and consequent approximation of 
certain constituents from each pattern, an interaction occurs 
of one on the other. Such an effect can only be produced 
if the two patterns are produced in the same source. Hence 
the original components of such a doublet must be produced 
simultaneously in each atomic configuration, whether a 
magnetic field is present or not. It follows that in radiation 
there must be simultaneous passages of two electrons, each 
from its original orbit to its final one. But as the effect 
takes place at one operation, the total change of energy 
is passed on to the radiator and emitted as a single mono- 
chromatic radiation, i. e. no doublet. It might be suggested 
that the effect could be explained on the hypothesis that the 
magnetic field affects the mutual possible orbits, and that 
sometimes one passage occurs and sometimes the other. It 
is difficult to see, however, how an orbit can be modified by 
another supposed one which is non-existent, i. e. not being- 
described at the same time, 

4. This consideration does not affect evidence for the 
quantum theory, but will serve to illustrate a habit which is 
somewhat exasperating in reading the writings of many 
exponents of the quantum theory — viz., the picking up of 
small and often irrelevant points as charming results of the 
theory. On p. 300 ff. it is expected that each doublet 
separation on passage from arc to spark conditions should 
be magnified in a measure corresponding to the ratio 4N:N, 
and satisfaction is expressed that in data adduced from 
corresponding elements in the doublets of group I. and the 
enhanced doublets of group II. this expectation is fulfilled. 
The ratios of the separations are reproduced (with Hg : Au 
added) in the first line of the following : — 

Mg. Ca." Sr. Ba. - 

5-3 3-9 3-4 3-1 

2-24 210 2-03 1-92 

* 'Analysis of Spc^tr.i/ § 7, p. 96. 



Zn. 
3-5 


ca. 

2-7 


Eu. 

9 


2-5 


2-21 


242 


2-03 


0.]9 



Quantum^ Orbit Theory of Spectra. 351 

But surely these numbers show that the comparison is not 
justified. As is known, a correspondence actually lies 
between the enhanced doublets and the triplets in the 
same element. Thus the ratios of the doublets to the first 
separation of the triplets are given in the second line of 
figures above, where the agreement is remarkably close. 
In this latter case, however, the correlation is not direct. 
It is due to three concurrent facts : (1) ratio 4N : N ; 
(2) the oun multiples in the doublets and v\ of the triplets 
are very nearly the same in each element ; and (3) the 
denominators in the doublets and triplets have nearly the 
same ratio in all (see belowj. There is, however, a close 
correspondence between the mantissse of the doublets in the 
two groups I. and II., especially as between the alkalies and 
the alkaline earths, those of the latter being about double the 
first. Correlation is also shown between" the denominators 
of the triplet and doublet sets in all the group II. elements. 
These statements are illustrated by the following data : — 

II. I I. 





Tripl. 


Doubl. 


Ratio. 




Doubl. 


Ratio 


Mg ... 


... 1-660 


2-265 


1-36 


Na .... 


. 2-117 


2-27 


Ca ... 


... 1-796 


2498 


1-39 


K 


. 2-235 


2-12 


Sr ... 


... 1-880 


2611 


1-39 


Kb .... 


. 2-292 


2-09 


Ba ... 


... 1-957 


2735 


1-39 


Cs .... 


. 2-361 


2-03 


Zn- ... 


... 1-599 


2-098 


1-31 • 


Cu .... 


. 1-869 




Cd ... 


... 1641 


2-144 


1-30 


Ag -. 


. 1-892 




Eu ... 


.. 1-648 


2190 


1-32 









Hs... 


.. 1-653 


? 




Au .... 


. 1-929 





Here 'under II. the third column gives the ratio of the 
denominators ; under 1. the second column gives tlie ratio oj 
mantissa' in JLl. to those in I. 

On the other hand, there appears very little correlation 
between the oun multiples which give ,the separations in 
corresponding elements of groups I. and II. A much 
closer one is found between those of the triplets and doublets 
of the same element in II. Thus, in the following, the first 
line gives the ratios of the oun multiples of doublets in 
group I. to those of the enhanced doublet of the corre- 
sponding element in group II. The second line gives the 
ratios of these multiples for the first triplet separation and 
the enhanced doublet in each element. 



Mg. 


Ca. 


Sr. 


Ba. 


Za. 


Cd. 


•683 


•779 


•843 


•903 


•847 


11 


•391 


•709 


•726 


•761 


•789 


•846 



Eu. 



•842 



He-. 



[ 352 ] 

XXXI. On the Theory of the Characteristic Curve of a Photo- 
graphic Emulsion. (Communication No. 22 from the 
British Photographic Research Association Laboratory.) 
By F. ( !. Toy, M.Sc, F.Inst.P., F.R.P.S. * 

IN the most recent investigations on the relation between 
the photographic effect and the light-exposure, special 
plates containing only a single layer of grains have usually 
been employed. With such plates the photographic effect 
is determined by counting the percentage of grains made 
developable. The curve expressing the relation between this 
percentage (x) and the logarithm of the exposure may be 
called the characteristic curve of a single-layer emulsion, 
corresponding to the ordinary curve of a commercial 
emulsion, in which, instead of x, values of the density (in the 
photographic sense) are plotted. 

In a recent paper (Phot. Jour. 1921, lxi. p. 417) the author 
has shown that such a curve, for a set of grains which are 
geometrically identical, is of the usual S-shaped tj^pe, i. e. a 
difference in size or shape does not account for the fact that 
all the grains do not become developable with the same ex- 
posure. Now, a set of geometrically identical grains, all in 
a single layer and similarly orientated to the incident light, 
represents the simplest possible emulsion which we can in- 
vestigate experimentally. It also corresponds to the simplest 
theoretical case, eliminating many complicating factors 
which, though greatly affecting the form of the characteristic 
curve, have nothing to do with the primary mechanism of 
the photograph process. In other words, with this emulsion 
the curve is reduced to its " purest " form, and is determined 
almost solely by the photochemical process which takes 
place. 

It is now generally believed that the primary action of 
light on the grains is to form in or on the surfaces of them 
certain " centres " or " points of infection " which act as 
starting-points for their reduction by the developer. This 
view has for some time had considerable evidence in its 
favour. Chapman Jones (Phot. Jour. 1911, li. p. 159) 
showed that by stopping development at a very early stage 
it is possible to get particles of silver too small to be visible 
microscopically, but which can be shown to be present by the 
colour imparted to the film, and by enlargement to visible 
dimensions by the deposition on them of mercury. Hodgson 
(Brit. Jour. Phot. 1917, p. 532) carried development a little 

* Communicated by Prof. A. W. Porter, F.R.S. 



Characteristic Curve of a Photographic Emulsion. 353 

further, and showed it possible to observe the silver reduced 
by the developer only around certain centres in the grain, 
and a recent paper of Svedberg's (Phot. Jour. April 1922) 
leaves little room for doubt that the possibility of a grain 
being made developable depends on the existence in it of 
some kind of reduction centre. 

Opinion as to the nature of these centres seems at present 
to be divided. There are those who assert that they are 
formed by the light-action, and that they do not exist before 
exposure is made. Such, for example, is the case if the 
centre is really a molecule of silver halide which has lost an 
electron, as is believed by H. S. Allen (Phot. Journ. 1914, 
liv. p. 175). On the other hand, there are those who believe 
that the centres are actual particles other than silver halide 
formed in the grains during precipitation and subsequent 
ripening, and that these only become susceptible to the action 
of developer after exposure to light. 

There certainly is considerable evidence to show that 
silver halide is not the only substance in the grains. Luppo- 
Oramer (Kolloidchemie una 1 Photographie) was led, as a result 
of his work, to the conclusion that, at any rate in the most 
sensitive emulsions, nuclei are present which probably consist 
of a colloidal solution of silver in the halide. Renwick 
(J. S. C. I. 1920, xxxix. No. 12, 156 T.) extends this idea, 
and says : " In our most highly sensitive photographic plates 
we are dealing with crystalline silver bromide in which, 
besides gelatin, some highly unstable form of colloidal silver 
exists in solid solution, and it is this dissolved silver which 
first undergoes change on exposure to light." These silver 
particles are negatively charged, and Renwick believes that 
the action of light is to discharge, and hence to coagulate 
into larger groups, those particles of colloidal silver which 
existed in the grain before exposure ; it is these groups of 
coagulated electrically neutral particles which are the re- 
duction centres. This view is supported by the ultra-micro- 
scopic observations of Gralecki {Koll. Zeit. 1912, x. pp. 149- 
150), who showed that X-rays have a coagulating effect on 
the particles in gold sols ; by Svedberg {Koll. Zeit. 1909, 
iv. p. 238), who has similarly shown that ultra-violet light 
agglomerates ultra-microns to larger aggregates; by Spear, 
Jones, Neave, and Shlager (J. Ainer. Chem. Soc. 1921, xliii. 
p. 1385), who have observed the same kind of effect with 
colloidal platinum; and by recent experiments of Weiger 
and Scholler {Sitz. Preuss. Akad. Wiss. Berlin, 1921, 
pp. 641-650). 

PA/7. Maq. S. 6. Vol. 44. No. 260. Aug. 1922. 2 A 



354 Mr. F. C. Toy on the Theory of the 

These facts are at any rate sufficient to justify an attempt 
to explain the relation of the number of grains changed to 
the light-intensity on the basis of the existence in the 
grains of: actual particles which are not silver halide. These 
are not necessarily all changed to reduction centres from in- 
active particles by the same light-energy as they would be 
if they were single molecules of the halide. We shall make 
no assumptions as to the composition of these centres, and 
the theory does not depend on their being composed of 
colloidal silver. We shall use the term " nucleus " rather 
than centre to indicate the presence in the grains of actual 
particles before exposure. 

Ihe Characteristic Curve of a Set of Geometrically 
Identical Grains. 

llieoretical. 

Our first object is to consider the case of a set of grains of 
identical size and shape, and to determine the relation we 
should expect to find between the percentage of these which 
are made developable and the light-intensity. The time of 
exposure is kept constant throughout. 

If we consider a volume V of the silver halide which is 
very large compared with that of a single grain, we may 
assume that the total number of nuclei in any such volume 
of the emulsion is the same, though the number contained in 
individual grains in this volume may vary. We will define 
the sensitivity of a single nucleus as the minimum intensity 
which must be incident upon it in order to make it " active " 
in the presence of the developer. For a given intensity of 
the incident light there will be a definite number of such 
active nuclei in every volume V, and they will be distributed 
amongst the grains entirely haphazard, according to the laws 
of chance. Every grain which happens to have at least one 
active nucleus will be developable. 

When the intensity of the light increases, more grains are 
changed. On any " nucleus " theory this happens because. 
more nuclei are present, so that a single grain has a greater 
chance of having at least one of them. This may be ex- 
plained in one of two ways. Firstly, all nuclei may have the 
same sensitivity, say I, but owing to the rapid absorption of 
light, those nuclei which are situated in the grain at some 
distance from the surface on which the light is incident, do 
not receive an intensity of I when the incident intensity is 
small. As the latter becomes greater, the volume of silver 
halide, throughout which the intensity is at least I, increases, 



Characteristic Curve of a Photographic Emulsion. 355 

so that the number of active nuclei increases also. Secondly, 
the sensitivity of every nucleus may not be the same, so that 
as the intensity of the light is increased, nuclei become 
operative which are unaffected by lower intensities, and 
again the total number of active nuclei increases with the 
intensity. 

We will consider only the case of grains in the form of 
thin plates as they occur in high-speed emulsions. Eggert 
and Noddack (Preuss. Akad. Wiss. Berlin. Ber. 1921, xxxix. 
p. 631) have recently measured photometrically the fraction 
of the incident light which is absorbed by an ordinary 
commercial photographic plate, and have found it to vary 
with the different plates from about 4 to 12 per cent, for 
violet light, for which the amount of light absorbed is near 
the maximum. Now, these plates contain several layers of 
grains, so that a very extreme upper limit to the fraction of 
light absorbed by a single grain is, say, 20 per cent. Thus, if 
there is an increase in the incident intensity of the order of 
20 per cent., the intensity of the light transmitted through a 
grain will be equal to the intensity incident before the 
increase took place. Thus, if all nuclei are equally sensitive, 
a change in the incident intensity of the order of 20 per 
cent, will cause a difference in the number of active nuclei 
from zero to some fixed maximum, so that the characteristic 
curve can only function over a range of intensity such that 
the ratio of its extremes is of the order of 1*2 : 1. As will 
be shown later, for the steepest characteristic curve plotted 
this ratio is about 25 times as much as this, so that as an 
appreciable factor in determining the increase of nuclei 
with intensity the first assumption is untenable. We have, 
therefore, to assume that all the nuclei are not equally 
sensitive. 

Since these nuclei are all formed in the same emulsion, 
most of them will have a sensitivity near the average value 
for the whole, and there will be a few which are very 
sensitive and a few which are very insensitive. There will 
be none which will respond to zero intensity, and none so in- 
sensitive that it takes an infinite intensity to affect them. 
We therefore expect the curve showing the relative number 
of nuclei R having any given sensitivity I to be of the 
general form shown in fig. 1. The exact mathematical form 
of this curve is immaterial at present, but it will be similar 
in general form to that obtained by Clerk Maxwell for the 
distribution of velocities between the molecules of a gas. By 
similar reasoning to his, the number of nuclei (Nj) which 

2 A2 



356 Mr. F. C. Toy on the Theory fo the 

have sensitivities between zero and I x (which is the number 
operative when the intensity of the light is Ij) is given by 

Fig. 1. 




o i, 

the area OAB, i. e. 



i-» 



N,: 



J/ a)dI ' 



(1) 



where /(I) gives the values of the ordinates in terms of I ; or 
if N is the average number of nuclei per grain and a the 
number of grains in volume V, 



N, 



■-;J>.*. . . , 



(2) 



The total number of nuclei is given by 
N=f/(I).c*I. 



Fig. 2. 




The curve showing the relation between N and I, shown in 
fig. 2, is characterized by its unsymmetrical-shaped S form. 



Characteristic Curve of a Photo<jrapliic Emulsion. 357 

When the a grains in the volume V of silver halide are 
subjected to an intensity I t , every grain which happens to 
have at least one of these N : nuclei will be made developable. 
We have, therefore, to find the chance of a grain containing 
at least one of the N\ nuclei when they are distributed hap- 
hazard amongst a grains. This can easily be obtained from 
the theory of probability. 

If p denotes the very small probability that an event will 
happen on a single trial, the probability P r that it will happen 
r times in a very great number, say n trials, is (Mellor, 
1 Higher Mathematics/ p. 502) 

V r =(n P y.e-»P/rl (3) 

Let the volume of a single grain be v, then since the volume 
of every grain is the same the total volume V is an. Let p 
be the very small probability that a volume dv will contain a 
nucleus, then 

p = ^ 1 .dvlav (4) 

To obtain the probability of the volume v containing a nucleus, 
we may suppose each dv to be a trial, so that the number of 
trials n is 

n = vjdv (5) 

Therefore the value of np in equation (3) is Nj/a, which is 
equal to N . If in this number of trials the event (i. e. v 
containing a nucleus) happens once, a grain will contain one 
nucleus; if it happens r times it will contain r nuclei, so 
that from (3), (4), and (5) we see that the probability of a 
grain containing r nuclei is 

p r =(N y*-^/H, (6) 

which is the same equation as was obtained independently 
and first published by Svedberg. The probability of a grain 
containing no nuclei is the value of this expression when 
r = ; i. e. 



Now, since it is certain that a grain must contain either zero 
or at least one nucleus, the probability P x that a grain will 
have at least one is 

P 1 = 1_,-n (7) 



358 Mr. F. C. Toy on the Theory of the 

But if x\a is the fraction of grains which are changed, 
F 1 = xja ; 

or, denoting log a/ (a — x) by A, we have 

A = N . 

Thus the same form of curve should be obtained when N is 
plotted against the intensity as is obtained when A is plotted 
against the same variable. The form of this curve should 
be an unsymmetrical S, as shown in fig. 2. 

Experimental. 

The first experiment carried out was to determine the 
relation between A and the light-intensity I for a set of 
geometrically identical grains, every grain counted being 
measured, as described in a previous paper (Phot. Jour. 
192 L, lxi. p. 417). 

Table I. 

Cross-section of grain = 0*98/x, 2 . 



Log I 

(relative). 


X. 


X 

(curve- 
values.) 


A. 


Log I. 


X. 


X 

(curve 

values.) 


A. 





92-0 


91-5 


2-45 


-0-893 


47-7 


466 


0-62 


-0162 


89-5 


90-0 


2-30 


-1-215 


5-2 


5-2 


0-05 


-0-310 


87-7 


87-0 


2-04 


-1-487 


o-o 


o-o 


o-oo 


-0-572 


74-8 


750 


1-39 


-1-788 


00 


o-o 


0-00 



In the first two columns of Table I. are shown the values 
of log I and x determined experimentally by exposure behind 
a neutral wedge, and these are plotted in fig. 3. The values 
of x given in column 3 are read off the curve in fig. 3, and it 
is these values which are used in calculating A in column 4. 
This is the best way of obtaining the A values, since when x 
is large a very small error in its determination means a very 
large error in A. The A, I curve, shown by the solid line 
in fig. 4, is exactly as predicted by the theory. 

We must note here that this is not in agreement with the 
results of Slade and Higson (Proc. Roy. Soc. 1921, A, xcviii. 
p. 154) and a previous experiment of the author (ibid. 1921, 
c. p. 109) to confirm their result. Slade and Higson stated 
that the relation between A and I can be expressed by the 
equation 

Arrairi-^ 1 ), 



Characteristic Curve of a Photographic Emulsion, 359 

where a and ft are constants. A comparison of the form of 
this curve (fig. 5) and the curve in fig. 4 shows a difference 

Fiff. 3. 




i 

A 



► T 










O 




/ 


-" 




? 


/ 
/ 

/ 










/ 
/ / 








1 










I / 










/ 











0-2 04 0-fc 0-8 1-0 



360 



Mr. F. C. Toy on the Theory oj the 



when I is large, but this can be explained. Firstly, the 
grains used in Slade and Higson's experiments were not all 
of one size, the variation being about 30 times that in the 
present case. Secondly, the best curve given in Slade and 
Higson's paper has actually the same form as that in fig. 4 
if equal weight is given to each point plotted. Also, in the 
author's confirmatory experiment, the main point was to show 
that at low intensities A varied, at any rate approximately, 
as I 2 ; at high intensities the work was not nearly as accurate 

Fig. 5. 




I— > 

as in the present case. To be certain of the form of the 
curve in fig. 4 the upper part of it was plotted for another 
size of grain, and that the same result was obtained is shown 
by the dotted line. 

The position of a nucleus can be detected by Hodgson's 
method of partial development of the exposed grains (ibid.). 
The developer used was made up as follows : — 

200 c.c. saturated Na 2 S0 3 , 
8 c.c. 10 per cent. KBr, 
0*3 gm. Amidol. 

This is a weak, slow developer, and is best for this purpose 
because there is a bigger latitude than if the developer is 
strong in the time of development necessary to render the 
position of the nuclei visible and yet distinct from one 
another. The best development time was found by trial and 
examination of the grains under the microscope. After 
exposure the plate was plunged into the developer for a 
known time, then quickly and thoroughly washed, and dried 
without fixing. The flat triangular grains used were so thin 
that the silver deposit was visible without dissolving away 



Characteristic Curve of a Photographic Emulsion. 361 

the silver bromide. In fig. G are given some examples of 
grains in which these nuclei appear ; they are formed more 
on the edges of the grains than anywhere else, though quite 
a number appear either inside or on the flat surfaces. 



Fig-. 6. 




Positions of nuclei numbered. 

The next experiment was to show if equation (6) holds 
good. A plate was exposed to a uniform intensity, partially 
developed, and the number of nuclei occuring on each of 
150 grains was counted. Hence the average number per 
grain was known and also the number of grains having 0, 1, 
2, 3, etc. nuclei each. 

In Table II. are given the theoretical and observed values 
of P,. for two equal-sized sets of grains, in one and the same 
emulsion, having widely different values of N . 



Table II. 



N = 0-180. 


N = 1-193. 


Value 
of r. 


No. grains 
having r 
nuclei. 


Probs 

Obs. 


ability 
Pr. 
Calc. 


Value 

i of r. 


No. grains 
having r 
nuclei. 


Probability 
Obs. Calc. 





91 


O607 


0619 


i 


43 


0-287 0303 


I 


47 


0313 


0297 


1 


55 


0-367 0362 


2 


11 


0073 


0071 


2 


36 


0240 0-216 


3 


1 


0007 


0001 


3 


12 


0-080 0-086 


4 





0000 


0001 


4 


4 


0-027 0-026 



The observed values of P r were determined by the fact that 
the probability of a grain having r nuclei is equal to the 
fraction obtained by dividing the number of grains which 
have r nuclei by the total number, i.e. 150. In fig. 7 the 
theoretical values are represented by the smooth curves, and 



362 Mr. F. C. Toy on the Theory of the 

those observed by the plotted points. The agreement is very 
good, and proves the validity o£ equation (6) in the case of a 
fast emulsion. 



Ker. 7. 



0-8 



Ob 



r ro-4 



0-2 















1 

1 






>< 








10 N = 0-480. 
W N Q =1-H3. 




\ 








\ 


\ 




















^ 


K 
















u 




\ 




















- \ 


\ 


X 


^) 






































> 




i 

r 


__J- 


5 



To find the relation between the average number of nuclei 
per grain and the intensity, a plate was exposed behind a 
step wedge and partially developed. The size of grain 
selected was the same as used for determining the A, I curve 
in fig. 4, the plate being exposed for approximately the same 
time behind the same wedge. At each intensity (I) the total 
number of nuclei on 200 grains was counted (except at 
1 = 0*044, where 100 grains were considered sufficient), and 
hence the average number per grain found. The values are 
given in Table III., and it will be seen that the curve in 
fig. 8 is of the same general form as the A, I curve in fig. 4, 
as is predicted by the theory. 

Table III. 
Cross-section of grain = 0'98/u, 2 . 



I. 


N . 


L 


N . 


1-000 


0-98 


0-270 


0-24 


0-689 


0-88 


0-180 


005 


0-490 


0-63 


0-128 


0-02 


0-356 


43 


0-044 


o-oo 



Characteristic Curve of a Photographic Emulsion, 303 

The highest value of N is about 1, which corresponds to 
less than 70 per cent, of grains changed, whereas actually 
the percentage changed corresponding to this value of N was 
about 90. Tbis is because the partial development has not 
been sufficient to show up all the nuclei, and it is very 
difficult to do this, since before this stage is reached, nuclei 
which initially were distinguishable from one another have 
grown together into a single mass of silver. It is, however, 
very unlikely that even if every nucleus could be observed 
the general form of the curve in fig. 8 would be changed. 



Fte. 8. 



+00 



N, 



O QSfr 



op 



025- 



-e- 




0-2S 



0-50 



0-75 



■00 



The most natural assumption to make is that longer develop- 
ment would merely result in an increase of the number of 
visible nuclei in proportion to the number already observable, 
and that this is the case is shown by the following experi- 
ment : — Two plates were given the same exposure under the 
wedge and partially developed, one for 15, and the other for 
18 seconds. The values of N were then found for widely 
different intensities, with the following results : — 

(1) I = 1*00, N for 15 seconds development = 0*613, 
N for 18 seconds = 0'980, whence (N ) 18/(N )15= 1-59. 

(2) I = 0*27, (N )15 = 0-153, (N )18 = 0*240, whence 

(N ) 18/(N ) 15 = 1*57 ; so that this ratio is practically 
constant, and the general form of the curve is indepen- 
dent of the development. 



364 



Mr. F. C. Toy on the Theory of the 



Variation of Grain Size. 

Experimental. 

It will be convenient to deal first with the experimental 
curves. When the values of x were being found for the 
curve in fig. 4, the corresponding values for three larger 
sizes of grain were determined at the same time and in the 
same way. The characteristic carves for the four sizes are 
shown in fig. 9. 

Fig. 9. 




Logl-> 



The important points in regard to these curves are that for 
one and the same emulsion : — (1) a set of large grains is 
more sensitive than a set of small ones, which confirms 
Svedberg and Anderson's result (Phot. Jour. 1921, Ixi. 
p. 325) ; (2) the characteristic curve for small grains has a 
greater maximum slope than that for large ones, i. e. } the 
ratio of the intensity which just changes all the grains to 
that which just causes the smallest possible change is larger 
the larger the grain size. As will be seen from the figure, 
the logarithm of this ratio for the smallest size grain is about 
1'5, which is equal to an intensity ratio of 30 : 1, whilst for 
the largest size a ratio of 100 : 1 is necessary to give half the 
curve. 

In Table IV. are given values of x, as read off the experi- 



Characteristic Curve of a Photographic Emulsion. 365 

mental curves in fig. 9, corresponding to known relative 
intensities, and in the third column the values of: A are 

Table IV. 
Cross-section of (a) =0'98yu, 2 , 
(6) = 1-75 M 2 , 
(c)=2-73/, 

(<0-8-9V- 





X 


(curve 


values) . 




A. 




Relative 
Intensity. 


































1-000 


(a). 


(ft). 


(«). 


(d). 


(a). 


(b). 


(')■ 


(<*)■ 


91-5 


95-8 


97-5 


98-0 


2-45 


3-16 


3-70 


391 


0-689 


900 


94-5 


96-8 


97-8 


2-30 


2-91 


3-45 


3-80 


0-490 


87-0 


93-0 


96-0 


97-0 


2-04 


2-66 


3-22 


3-51 


0270 


75-0 


87-6 


92-5 


94-5 


1-39 


2-08 


2-60 


2-90 


0-128 


460 


73-0 


81-0 


88-5 


0-62 


1-31 


1-66 


2-16 


0-06 I 


5-2 


52-5 


66-6 


78-0 


0-05 


0-74 


1-10 


1-51 


0-033 


00 


36-0 


54-0 


67-6 


o-oo 


0-45 


0-78 


1-12 


0016 


00 


20-7 


40-5 


570 


o-oo 


022 


0-52 


0-84 


0-008 


o-o 


70 


28-5 


46-5 


o-oo 


0-07 


0-33 


0-62 



Fig. 10. 



T 
A* 





s / 






// 


/ / 


^^T 

^^^r 




t 

















0-25 



0-50 



075 

X- 



1-0 



calculated. The A, I curves for the four sizes of grains are 



given in fig. 10. 
© © 



366 



Mr. F. C. Toy on the Theory of the 
Iheoretical. 



Consider what is the effect of a variation in grain size on 
the nuclei distribution curve shown in fig. 1. 

We will first assume that the sensitivity of a nucleus is 
quite independent of the size of the grain in which it chances 
to be, i. e. once a nucleus is formed in a grain, its sensitivity 
does not change as the grain grows. This is apparently 
Svedberg's assumption, for he says : " the small and the larger 
grains in one and the same emulsion are built up of the same 
kind of light-sensitive material — -just as if they were frag- 
ments of different size from one homogeneous silver bromide 
crystal/' If this is the case, then the only result of in- 
creasing the size of grain is to increase the total number of 
nuclei, and these will be distributed amongst the different 
sensitivities in the same proportion as before. This is shown 
in fig. 11, where the distribution curves for two sizes of 

Kff. 11. 




grain are given. We have made no assumption regarding 
the relation between total number of nuclei and grain size 
except that large grains have more than small ones. 

The curves relating I and N (average number of nuclei 
per grain) which will be obtained from distribution curves 
such as those in fig. 11 are shown in fig. 12. We have already 
shown that the N , 1 curve is identical in form with the A, I 
curve, so that those in figs. 12 and 10 should be of the same 
form. As a matter of fact, there is a striking difference. 
The experimental curves in fig. 10 lie practically parallel to 
one another at the higher intensities, and the point of in- 
flexion (which corresponds to the maximum ordinate in the 
nuclei distribution curves in fig. 11) moves towards the origin 
as the grain size increases. In curves (b), (c), and (d), 
fig. 10, which are for exceedingly sensitive grains, the point 



Characteristic Curve of a Photographic Emulsion. 367 

of inflexion has moved so near the origin that the part of the 
curve to the left of this point does not show on the scale to 
which the curves are plotted. On the other hand, the 
theoretical curves in fig. 12 are characterized by the fact 

Fig. 12. 




that the ratio of the ordinates for different sizes of grain is 
independent of the intensity, and the value of I at the points 
of inflexion, 1; and the average sensitivity do not change as 
the grain size is varied. Thus we cannot explain the effect 
of a variation of grain size on Svedberg's assumption. 

Xow let us assume that the sensitivity of a nucleus depends 
on the size of the grain in which it is contained, and that if 

Fig-. 13. 




it is in a large grain it is more sensitive than it would have 
been in a small one. The effect of this on the distribution 
curve for the larger grain in fig. 11 is to shift it bodily 
nearer the zero, thus decreasing the value of I; and increasing 
the average sensitivity, as in fig. 13. The N I curves 



368 



Mr. F. C. Toy on the Theory of the 



plotted from these distribution curves are shown in fig. 14, 
and it will be seen that they are similar to the experimental 
curves in fig. 10. The reason why, for very sensitive grains, 
the lower half of the 8-shaped curve appears to vanish (b), 
(c), and (d), fig. 10, is that the value of Ij is very nearly 
zero, but it would be shown if the points were plotted on a 
bigger scale. 

Fig. 14. 




The evidence thus points to there being two reasons why 
large grains are more sensitive than small ones. Firstly, 
there are more nuclei present in the larger grains, so that a 
single grain has a greater chance of having at least one ; and 
secondly, the average sensitivity of the nuclei increases with 
the size of grain. 

Svedberg in his most recent paper (ibid.) discusses the 
relation between the average number of nuclei per grain and 
the grain size. He says: — " The rapidity of the increase of 
the average number of nuclei per grain N with size of grain 
would depend on two factors : the ability of the developer 
to penetrate into the grain, and the homogeneity of the field 
of light in the grain. If the developer is not able to get 
into the interior of the grain, but only attacks the surface 
layer, then N would mean the number of centres in that 
surface layer, and therefore would increase in approximate 
proportion to the grain surface even in cases where the field 
of light in the grain was not homogeneous (because of strong- 
light absorption). On the other hand, if the developer is to 
penetrate the grain, N~ would depend upon the field of light 
in the grain. If the absorption of light were feeble, N would 
increase in proportion to the volume of the grain ; if the 
absorption were very strong, N would increase approximately 
proportionally to the cross-section of the grain/' Later in 



Characteristic Curve of a Plwtograpliic Emulsion. 360 

the paper he compares the variation of N with grain size 
for grains which have been exposed to light with the variation 
when the exposure is to X-rays, and suggests certain deduc- 
tions as regards the absorption of light and X-rays by the 
silver halide from the difference which he finds. 

Now, from fig. 10 we see (since A=N ) that the manner 
in which N varies with grain size depends on the intensity 
to which the grains have been exposed ; we can select an 
intensity such that N varies in almost any manner we please. 
Thus, unless the difference between Sved berg's results and 
those found here is due to the different emulsion used, there 
seems to be no justification for making deductions from the 
relation which is found between N and the size of grain at 
one fixed arbitrary exposure. 

The theory which has been advanced here is capable of 
explaining an important fact which appears quite inexplicable 
on such a theory as Allen's (ibid.). It is well known that the 
sensitivities of the grains in an emulsion depend to a great 
extent on the conditions of precipitation and ripening; and 
that, in different emulsions, sets of equal-sized grains may 
have quite different sensitivities, and even different maximum 
slopes for their characteristic curves. If, as Allen suggests, 
the nucleus is really a simple molecule of silver halide which 
has lost an electron, its characteristics will be the same 
whatever the emulsion, and it is difficult to see why grains in 
one emulsion should be more sensitive than those of the same 
size in any other emulsion. If, however, the nucleus is not 
silver halide, it is very probable that the conditions of pre- 
cipitation and ripening do play an important part in deter- 
mining its characteristics. 

Thus, on Ren wick's theory, the condition of the colloidal 
silver which is produced will certainly depend on such factors 
as the kind of gelatin, conditions and time of ripening, etc., 
and the ease with which colloidal silver particles can be 
coagulated will be affected by the amount of gelatin present, 
since this is a protective colloid. The great difficulty 
in accepting Renwick's theory as it stands is this : — It is 
known that an unprotected silver sol is very stable to the 
action of light. Therefore, if a protective colloid is present, 
it will be still more difficult to effect its coagulation and 
precipitation by light, whereas in the case of our most 
sensitive silver halide grains the energy necessary to make 
them developable is exceedingly small. 

Luppo-Cramer (ibid.) believes that the mechanism of the 
formation of the latent image is not the same for the most 
sensitive and very insensitive emulsions, and he claims that 
Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. 2 B 



370 Characteristic Curve of a Photographic Emulsion. 

this is supported by his experiments. He found that the 
sensitivity of a very fast emulsion was decreased considerably 
by treatment with chromic acid, but that the sensitivity of a 
very slow emulsion remained unchanged. He explained 
this by the existence on the surface of the sensitive grains of 
colloidal silver, formed during the ripening process, which 
was not present in the insensitive grains, and which was 
removed by the chromic acid. 

It is very difficult to imagine that the fundamental light 
action varies with the kind of emulsion, and that considering 
a whole series of emulsions, from the most sensitive to the 
most insensitive, there is a transition region where an entire 
change of mechanism takes place* Strong evidence against 
Liippo-Cramer\s view is that Svedberg (ibid.) has shown that 
in one of the slowest emulsions the reduction centres are 
distributed amongst the different grains according to the 
same law as has been shown here to hold for their distribution 
in the case of one of the fastest commercial emulsions. 
This is in favour of the view that for all kinds of emulsions 
the process of the formation of the latent image is the same. 

The existence of this chance distribution of developable 
" centres " in the grains does not conclusively prove that 
they are the kind we have considered in this paper, and there 
are at least three other possibilities. Assuming a discrete 
structure of the radiation, the centres may, as suggested by 
some, be the points of impact of light quanta on the grains, 
but the fact that the majority of these centres are located on 
the edges of the grain is strongly against this view. Also 
within the crystal there maybe a chance concentration of the 
light energy at certain points, and both these possibilities are 
being tested in this laboratory. Again, this chance dis- 
tribution may be due merely to the fact that the grain as a 
whole is changed by the light, but the developer reaches 
some points of it sooner than others. If this is so, there 
appears to be no reason why the average number of centres 
per grain, considering only developable grains, should in- 
increase, as it does, in a regular manner with the light 
intensity. The author believes that the evidence so far 
obtained is mainly in support of the theory discussed in this 
paper. 

In conclusion, the author wishes to express his thanks to 
Dr. T. Slater Price, Director of Research of the British 
Photographic Research Association for much valuable 
criticism and advice. 



. On the Stark Effect for Strong Electric Fields. 371 

Summary. 

A theory is advanced which explains the relation found 
experimentally between the number of geometrically identical 
silver halide grains made developable and the light in- 
tensity. It is assumed that there exist in the grains particles 
which are not silver halide, and which are formed during 
precipitation and subsequent ripening. With any normal 
exposure (i. e. one which gives a value between and 100 
for the percentage of developable grains), it is these particles 
which form the reduction nuclei, the only action of the light 
being to change their condition in such a way that they 
become susceptible to the action of the developer. Each 
nucleus does not necessarily require the same intensity to 
change it. The nuclei are scattered haphazard amongst the 
grains according to the laws of chance, and only grains 
which have at least one will be developable. The sensitivity 
of a grain is the sensitivity of its most sensitive nucleus. 

The effect of a variation of grain size is explained, and it 
is shown that Svedberg's assumption regardiDg the similaritv 
of the light-sensitive material in large and small grains is 
not in agreement with the experimental facts in the case of a 
fast emulsion. 



M 



XXXII. On the Stark Ejfeet for Strong Electric Fields. 

To the Editors of the Philosophical Magazine. 
Gentlemen, — 

Y attention has been drawn to the results of experi- 



ments by Takamine and Kokubu * in which an effect 
of the nature indicated in a recently published paper t of 
mine was detected, namely, a shift of the central line in the 
perpendicular component of Hy in a strong electric field. 
Before comparing the experimental amount of this shift 
with the theoretical value it would have on the Quantum 
theory of spectral lines,, it is necessary, however, to point 
out a slip in my paper referred to above : thus on p. 945 
a term is missing from the value of: the contour integral (4), 
instead of (6) the full value should be 

f = B D /3B* \ 5BD*/ 7B»\^ 

* "The Effect of an Electric Field on the Spectrum Lines of 
Hydrogen," I'art III. Memoirs of the College of Science, Kyoto 
Imperial University, vol. iii. p. 271 (1919). 

t Phil. Mag. xliii. May 1922, p. 943 ; this will be referred to freely. 

2 B 2 



372 On the Stark Effect for Strong Electric Fields. 

Consequently the third term on the right-hand side of 
equation (2b) p. 948 should be 

where N' is now given by 

-15n 3 2 -21(n 2 -n 1 ) 2 ~6-—^ ) i -- L 
3 v 2 iy nx + wa + rjj, j 

N(n) being still given by equation (10). In view of the 
identity 

N(n) = (2n 1 4-^3)(6n 2 2 4-6n^ 3 + n 3 2 ) 

+ (2?2 2 + W 3 ) (6?2 X 2 + 6 WjWg + Tig 2 ) 

= 3(n x + w 2 + ?2 3 ) 3 — 3(ni 4- n 2 + w 3 )(n 2 — n x ) 2 

— « 3 2 ( w i + w 2 + w 3 ), 
N' can be reduced to the form 

N'(n) = (n 1 + n 2 + n 3 ) 4 {17(n 1 + n 2 + w 3 ) 2 --9ri3 2 ---3(n 2 - y i 1 ) 2 } 5 

... (ii.) 

which shows in conjunction with (i.) that the remarks in 
the paper about the symmetry of the components are not 
affected by this correction. In order to calculate the amount 
of shift of the middle w-component of H y we observe that 
this component can arise from any of three possible transi- 
tions corresponding to 

(m 3 . m 2 , wii ; n 3 . n 2 , n ± ) 

= (3, 1, 1 ; 2, 0, 0) or (1, 2, 2 ; 2, 0, 0) or (1, 2, 2 ; 0, 1, 1) 

respectively, the values of {N'(n) — tN'(/n)} corresponding 
to these combinations being — 2'MxlO 5 , — 2*59 x 10 5 7 and 
again — 2*59 x 10 5 respectively. And on substituting the 
values of the universal constants in (i.) for hydrogen (E = e) 
the following expression is obtained for the wave-length 
shift 

\ 2 \ 2 F 2 



Damping Coefficients of Electric Circuits. 373 

This gives for H y (X = 4'34 x 10~ 5 ) and the value of F 
used by Takamine and Kokubu *, namely, 

F = 4-33 x 10 2 c.g.s. e.s. units [ = l'3x 10 5 volt x cm.- 1 ], 
AX, = *36A or *43A respectively. 

The experimental value observed by Takamine and Kokubu 
is about 1A, which is larger than that predicted by the 
Quantum theory. It is, however, possible that part of 
the experimental shift is due to a Doppler effect, and in 
any case the experiments could hardly be considered accurate 
enough to exclude o a possible experimental error of what is 
only about ^ an Angstrom unit. On the other hand, the 
photographs of the shift [plate ii. fig. 1] point decidedly to a 
(general displacement of all the components in the direction pre- 
dicted by the theory, namely towards the red, and may be 
taken as corroborative of at least this qualitative aspect of it. 
It is seen from (i.) and (ii.) that this lack of symmetry in a 
strong field would be expected on theoretical grounds to be 
more pronounced for the higher members of the Balmer 
Series {e.g. H$ or H 6 ), and it would be highly desirable to 
obtain measurements relating to these lines as a further test 
for the quantitative aspect of the theory. 

In conclusion I wish to thank Mr. W. E. Curtis for 

drawing my attention to the experimental results already 

referred to. 

T ^. , nil T q Yours faithfully, 

Kings College, London, J ' 

May 12th, 1922. A. M. MOSHARRAFA. 



XXXIII. On the Damping Coefficients of the Oscillations 
in Three- Coupled Electric Circuits. By E. Takagishi, 
Electro -technical Laboratory, Department of Communica- 
tions, Tokyo, Japan f. 

THOUGH the importance of the problem of three-coupled 
electric circuits has arisen with reference to radio- 
telegraphy, it does not seem to have been attacked with any 
great amount of attention except by B. Macku, E. Bellini, 
and very recently L. C.Jackson \. The valuable paper of 
the latter made me feel very much interested, especially as 
it will make an important contribution to radio fields, but 

* I. c. 

f Communicated bv the Author. 

X Phil. Mag. vol. xlii. No. 247, July 1921, p. 35. 



374 Mr. E. Takagishi on Damping Coefficients of 

unfortunately I found there slight errors concerning the 
damping coefficients of the circuits. 

Now, let us proceed to correct them, using the same 
notation and abbreviations for the sake of simplicity. 
Comparing coefficients in the . equations (5) and (6) in his 
original paper, we obtain, instead of (7), 

-2(q + r + s) 

= (R-iL 2 L 3 H- L 1 R 2 L 3 + L 1 L 2 R-3 

-(co^ + co^ + co/) 

_ J L^-M^ 2 L 2 L 3 -M 23 2 L 3 L 1 -M 31 2 

+ L 1 R 2 R 3 +RiL 2 R3 + R J R 2 L 3 \-r-D, . (ii.) 

-2{co 1 2 (r + s) + co 2 2 (s + q)+co 3 2 (q + r)} 

RJjg + LiR-a . R0X3 + L 2 R 3 R^ + LA 

C ^ d + ~cT 

+ R 1 R 2 R 3 |--D, . (iii.) 






C 2 C 3 C1O3 CiG 2 

RiR 2 R 2 R 3 R 3 Ri\ . -n r * 

2(&) 1 2 ft} 2 2 5-f- co 2 2 co 3 2 q -f cwg 2 ©! 2 **) 



•«i 2 (»2 2 a>3 2 



QAC 



-e-i), (tL; 



(7') 



where 

D = 2M 12 M 23 M 81 -L 1 L 2 L 3 

+ L 1 M 23 2 +L 2 M 31 2 + L 3 M 12 2 . (vii.) 

From equations (7') ii., iv., vi., vii. we obtain the same 
equation for co 2 as (8) and (9) in his original paper. For the 
damping coefficients q, r, and s, however, we find the 
following values, different from, those in (10). 



Oscillations in T 'hree- Coupled Electric Circuits. 375 

Now, making use of the abbreviations given below and in 
the original paper, we get the following equation : 

-2(?+ ;• + .<)=>, 

- 2 {./(or./ + O + K» 3 2 + °>i 2 ) + «(»i a + » 2 2 ) ! ■ = | , 



2f V 9 , 9 9. 9 



&)o-.^ = 



c 

X 



where 

A = * 1 (l-/9»)+*,(l-7»)-f*,(l -a 8 }, 

C = k^irn' 2 -{- Ii\ 2 P)i 2 + kj 2 m 2 , 

X = -(l-a 2 -/S 2 - 7 2 -2a^ 7 j. 

Solving these simultaneous equations, we have 

Q 

R 

'' = Y' 

S 
S =Y' 



in which 

A 
X 



Q = 



— 2 



v- — 2(w 3 2 + w 1 2 ) — 2(oj 1 2 + &) 2 2 ) 

C 
X 



!(&) 3 2 o) 1 2 ) — 2(o) 1 2 6) 2 2 ) 



A 1 1 

B (a)a 2 + ^ 2 ) K 2 + r, 2 2 ) 

/" ^ 9 9 2 9 



= |.(co 2 2 -a) 3 2 ) {A<-B Wl 2 + C}, 



similarly 



R = i (co 3 2 - Wl 2 ) {4« 2 4 -Bw 2 2 + C}, 



and 



Y - 8 (a>, 2 -G) 2 2 ) (of—uf) (a), 2 - *»/) 



376 Prof. S. C. Kar on the Eleetrodynamic 

That is, 

q = [^ 1 mV + ^n 2 /H/^H 2 )-o) 1 2 {/ 2 (/c 2 + ^) + m 2 (A-3 + ^ 1 ) 
4-n 2 (^ 1 -+^)+A ;i /cA} + a) 1 4 {^(l-/3 2 )+^(l- 7 2 ) 
+ /, 3 (1-^)}] 

- : - 2 (l_ a 2_£2_ 7 2_ 2a/ 5 7 ) (»*-**) (tof-CDs 2 ), 

r = [(/^ 2 +/^ 2 + y 2 Hi 2 )- w 2 2 { P fe + y -\-m 2 (h + h) 
+ n 2 ft^/. 2 ) + /^ 2 /^l + a, 2 4 {^(l-/3 2 )+A- 2 (l- 7 2 ) 
+ hil -a 2 )}] 

^2(l-a 2 -6 2 -7 2 -2a y S 7 )(a) 2 2 -a) 3 2 )(co 2 2 -a) 1 2 ), 

+ 72 2 ft + /v 2 ) + Ws} + o) 3 4 {^(l-^ 2 ) +/c 2 (l- 7 2 ) 

+ ^3(l-« 2 )}] 

^2(l-a 2 -/3 2 - 7 2 -2^ 7 ) (» 3 2 -ft>i 2 ) (a) 3 2 -0. 

On inspecting these equations for damping coefficients it 
is noticed, at once, they are also correct with respect to the 
dimensions. 



XXXIV. On the Eleetrodynamic Potentials of Moving 
Charges. By S. C. Kar, M.A., Professor of Mathe- 
matics, JBangabasi College, Calcutta *. 

THE eleetrodynamic potentials of a moving charge or 
the electron have been the subject of several in- 
vestigations and the earliest were those of Lienardf and 
WiechertJ. Among recent writers who have found the 
potentials on a relativity basis may be named Sommerfeld§ 
and M. X. Saha||. Both of these writers performed a four- 
dimensional integration in the Minkowski space-time mani- 
fold and have obtained results which are quite general. 
It appears to the present writer that the Lienard and Wie- 
chert result — and the method admits of easy extension to 
the case of a straight linear current — may be obtained easily 
enough by a Lorentz transformation to a rest-system and 
back without resort being had to four-dimensional integration. 

* Communicated by the Author. 

f JJ Eelairage electrique, vol. xvi. pp. 5, 53, 106 (1898). 

t Arch. Need. vol. v. p. 549 (1900). 

§ Ann. (I. Phys. vols. Hi. and liii. 

j| Phil. Mag. vol. xxxvii. p. 347 (1919). 



Potentials of Moving Charges. 377 

The equations for the potentials may be written 

□ (ic<&, F, G, H) = /3 /c(zc, u, r, ic) 

\ i / A a 2 + i' 2 H-«> 2 

where /c=1/a/ 1— 2 

It is well to point out that this mode of writing the 
equations is slightly different from the customary mode 

where QF-M so that our F is d¥'. This deviation 
c 

from usage is justified by the greater symmetry and homo- 
geneity of form resulting. The equations for h (magnetic 
intensity) and d (electric intensity) will on account of this 
•change assume the forms 

h=- rot(F, G, H), 

(2) 
It is evident that the operator □ is an invariant under 
n Lorentz transformation. It will therefore follow that 
(ic<&, F, Gr, H) is a four-vector, because ic(ic, u, v, iv) is a 
four-vector. Therefore c 2< &8t — FSx — G8y — HSr which 
represents the scalar product of the four-vectors (zc<l>, F, G,H) 
and (ic8t } 8x. hy, 8z) is invariant under a Lorentz trans- 
formation. Therefore, 

e 2 <P8t-F8x-G8y^R8z=c 2 &8t , --F'8x'--G'8y'--K'8z', 

where the dashes refer to a system of axes moving with 
velocity v along the axis of x. 

But 8x= K (8x' + v8t'), 
Sy = Sy, Sz = 8z', 

and bt = K ht' + =\ where k = l/.y/ 1 - % . 

Substituting and equating coefficients of 8x', 8y', Sz', and 
■8t' we have 

F'= K (F-v®), G' = G, H'=H, 

F 



and <&' = *(<I> — ^-Y 



378 Prof. S. C. Kar on the Electrodynamic 

These formulae are exactly similar to the usual formulae 
for Sx, 8y, Sz, 8t and connect the potentials for any system 
of axes with those of another moving with velocity v along 
the axis of x. 

The reversing formulse are 

F = *(F' + v<S>'), G = G', H = H', 
and ^> = /e /V+^. 

(3) 

Let us suppose an electron moving with velocity v 

along the axis of x and let us take a system of axes moving 

with the electron. It is apparent that for the latter system 

of axes the electron is at rest. The vector potentials 

(F', G', H') = and <!>' = - — , due to a static charge e where 

r' is the distance of the point P at which the potentials are 
considered. 

For the original system of axes, therefore, we should have 
according to the formulae of transformation given above, 

F=/™ * , G = 0, H = 0, and <£ = *«,—,. 

t is, however, expressed in terms of the coordinates of the 
rest system and it will be necessary to transform it to a form 
involving the coordinates of the original system. 

But if the time-difference between the point P and the 
electron is A*', then 

r' = cAt' 

- KG \ At 



r vAx~] r\ v Ax~\ 

— K\r = kt 1 . — 

L c J L c r J 



ev 



F = -,G = 0, H = 0, 



and <E> — 

4:TT7 



I" 1 --]' 

which are Lienard's results. 



Potentials of Moving Charges. 379' 

(4) 

Lei us suppose a straight linear uniform current to 
arise from continuous and uniform rush of electrons in the 
conducting wire in the direction of the current. Viewed 
from a system of axes moving with the common velocity of 
the electrons the phenomena reduce, as far as the rushing 
electrons are concerned, to the case of a linear and uniform 
distribution of electric charge. If N electrons each with 
charge — e be supposed to rush with velocity v to the 
observer in the rest-system the linear density of static charge 
is — N*. 

From the ordinary theory of potential, the potential <E>' for 
such a distribution is — 2N<? logV where r l2 =y' 2 + z' 2 and 
(F', G', H') = 0. Transforming to a moving system according 
to our formula? we should have 

F = Kv&=-2/eNev\ogr, G=Gr'=0, H = H' = 0, 

<5? = K & = -2/cNelogr [v r-=r']. 

The magnetic field therefore would be given by h x = 0, 

2xNevz 1 2/eNm/ , , . . „ , 
n v = 5 — , h = ^ —■ and the electric held would be 

, , A 7 2/cN^y , 2/cNez T ,, 
given by d x — 0, Oy^= y^ , a*= ^— . In the con- 
ducting wire, however, there is also a linear distribution of 
positive nuclei at rest of which the potential would be 
+ 2N<? log r. 

The electric field due to these would be given by d/ = 0, 
, mey ■ , M 2-Nez 

The resultant electric field would therefore have the 
components 0, !L ~V (1— /e), - — ~ (1 — k), and is of the order 

v 2 
of 2 . The magnetic field is of finite magnitude and cir- 
cular round the wire, the resultant being which is 

., . , . , . . 2 (current) .„ 

quite in accord with the expression it we put 

k±s ev iS ev 
current = or neglecting quantities of the order 

v 2 
2 in comparison with unity. 



[ 380 ] 



XXXY. The Identical Relations in Einstein' 's Theory. 
By A. E. Harward*. 

THE March number of the Philosophical Magazine con- 
tains an interesting proof of the identity 

by Dr. G. B. Jeffery. 

Apparently it is not generally known that this identity is 
& special case of a more general theorem which can be very 
easily proved. I discovered the general theorem for myself, 
but I can hardly believe that it has not been discovered 
before. 

The theorem is 

(BpyoPjr + (B^/)„ + (BptS)* =0. . . (1) 

This identity can be verified in a rather laborious manner 
by forming the covariant derivative of B^^P, but it can be 
more easily proved as follows : — 
The identity 

Aju, V( j — Aju, <Tv — j5fivo p Ap .... (2 J 

can be easily generalized so as to apply to the case where 
instead of the vector A^ we have a tensor of any order ; thus 

Aftv, or — Apy, tg == -t>ju<7r Apv ~T Dyur -^H-P' 

This is proved in the same way as (2). 
Now, if Afx be any covariant vector, then 

(Aju, vot — Aju, vt(t) -f (Aju, arv — &p., <rvr) + (^/z, rva A^ ; rav) 

= (A^, va — A^ ; cv)t-{- (A^, ar — Ap, rtrjv + (A^, rv — A^ V r)a\ 

J3 JLt(Tr |0 A.p^ v -\- D V(TT P Ap ( p -4 lJfiTV ^-p, <T + *5<TTV Ayu, p 
+ ^fxva P Ap, T + ¥>TV(T P Ap 5 p 

= (Bp Va P Ap)r + ( B^ T P A p ) v + (B Mr / Ap) a . 
* Communicated by the Author. 



The Identical Relations in Einstein s Theory, 381 
Now, 

(B^/ A p ) r = (IV/) r A p + BpvS A Pl r ; 

so after cancellation we get 

(B vff r p + B ( rr/4-Br V /)A /t p 

The expression in brackets on the left vanishes identically. 
Since Ap is arbitrary, the expression in brackets on the right 
must also vanish. Q.E.D. 
The identity 

B vaT P + B aTV P + B TV a<> = 

follows at once from the well-known identical relations 
between the Riemann symbols. The three-term identity is 
usually stated in the form 

(/uLTav) -f (/jlcvt) -f (/jlvtct) = 0, 

or in the modern notation 

Bfxpcrr~\~ ±>[xtv(t ~\~ tjjxarv == 5 

here B^cr denotes greB txva 6 = (firav), Since B [irv(T = 'B 1/(Tf j ir . 
and Bh(ttv = BcffivTi 

= B pV (JT 4" Bpa/JLT T j5<T/J.V7 

We assume that the determinant \g^v | = g does not vanish 
in the region under consideration ; therefore the expression 
in brackets must vanish. 

This identity can also be proved by observing that the 
expression 

(Av, or — Aj^, to) + (Aff ) tv — -A-ff, vt) 4" (A r, vff A-t, <jv) 

vanishes if A v is the derivative of a scalar; for in that case 

A Vt crr — A.cr,vT, Aa t Tv == At, <tv> and A^yu = Aj/,rcr. 

If we contract (1) by putting r = p, we get 



382 Mr. H. S. Rowell on Energy Partition 

If we contract this again by multiplying by gP v , we get the 
familiar identity 

<-H-?- ' (4) 

for since [gP v )p = 0, 
similarly 

and *r 

<f (G r , ff ) = (/» G„„)„ = (G)„ =g 

since G is a scalar. 

Jersey, 

13th May, 1922. 



XXXVI. Energy Partition in the Double Pendulum. 
By H. S. Rowell *. 

IN a letter to ' Nature ' (July 28, 1921) the present writer 
gave a theorem on the double pendulum which is capable 
of interesting extension. 

If the masses of the bobs are m and M and the respective 
amplitudes are a and A with suffixes. to denote the normal 
modes, then the theorem states that 

a Y a 2 M 

AjA 2 m ' 

If this equation is squared and both sides multiplied by 

m 2 /M 2 , we have 

ma^n 2 ma 2 2 n 2 2 _ 1 
— 1» 



MAxVMA/tis 

where nj and n 2 are the radian frequencies of the two moles. 
"This equation may be readily interpreted thus : — 

" The ratio of the kinetic energy of one bob to that of the 

* Communicated by the Author. 



in the Double Pendulum. 383 

other bob in one mode is the reciprocal of the corresponding 
ratio in the other mode/' 

Proceeding to the general case of an elastic system with 
two degrees of freedom, using Professor Lamb's notation, 

2T = A6 2 + 2U0cf> + B4>\ 

2 V = 2V + ad' 2 + 2h6<f> + b(f> 2 j 

so that with a time factor = in 

7i-(A0+H<£) = a6 + l«f>, 

?r(H<9 + B(£) = 7,0 + ty; 

whence the product of the roots in 6j^> is Ti — / & • 

If H = so that T is a function of squares of velocities, 
the product of the amplitude ratios is — B/A, or, in the 
double pendulum, — M/m. 

If A = so that the potential energy is a function of 
squares of displacements, the product of the amplitude 
ratios is bja, i. e. the ratio of the two stability coefficients. 
Thus in either case we have an energy relation. For the 
kinetic energy take 

H = and ££=-?; 

9192 A 

•square and multiply by A 2 /B 2 , and insert the frequencies. 
For the potential energy take 



A = and -V = — - 



0102 



i 
2 <* 



which, when squared as before, yields a similar relation. 
The two results may be expressed in words thus : — 

When the Kinetic or Potential Energy is written as a 
■function of squares only, the ratio of the Kinetic or 
Potential Energy expressed in one co-ordinate to that 
•expressed in the other co-ordinate for one normal mode is 
the reciprocal of the corresponding ratio for the other 
normal mode. 

This investigation gives an insight in certain cases into the 
indeterminateness of the normal modes with equal periods. 



[ 384 ] 

XXXVII. Velocity of Electrons in Gases. 
To the Editors of the Philosophical Magazine, 
Gentlemen, — 

IN a paper in the Jalirbuch der lladioactivitCd und Electronik 
(vol. xviii. p. 201, April 1922) H. F. Mayer gives an 
account of some of the formula? obtained by different 
physicists for the velocity of ions or electrons in gases due to 
an electric force, and concludes that a formula recently 
given by Lenard is more correct than the others. 

Among the other formulae which are discussed, the author 
gives what purports to be an account of a formula for the 
velocity of an ion which I published in the ' Proceedings of 
the Royal Society' (A. vol. lxxxvi. p. 197, 1912), and states 
that this formula is so incorrect that it does not even give 
the right order of the velocity. I should like to draw 
attention to the way in which Mayer has misinterpreted the 
matter, and to quote the formulae as I gave them for the 
different cases in which the mass of the ion is small or large 
compared with the mass of a molecule of the gas through 
which it moves. 

On pages 199, 204, and 206 of my paper, three formula? 
are given for the velocity U of an ion in the direction of the 
electric force X in terms of the mean free path I of the ion, 
its mass m, charge <?, and velocity of agitation u which is 
supposed to be uniform and large compared with U. 

The first of these is 

JJ = Xel/mu, (1) 

and applies only to cases in which the mass of the ion is 
small compared with that of a molecule of the gas (an electron 
for example), since it is here assumed that after a collision 
with a molecule all directions of motion of the ion are equally 
probable. 

I pointed out that when the mass of the ion is larger than 
that of a molecule of the gas, all directions of motion of the ion 
after a collision are not equally probable, and that in this 
case an ion travels a considerable distance (having an average 
value X) after a collision in the direction in which it was 
moving before a collision. A more general formula for the 
velocity was given, which is 

XJ = Xe(l+X)/mu. . . ! . . (2) 
If the mass m of the ion is so large compared with the 



Velocity of Electrons in Gases. 385 

mass m of a molecule of the gas that all directions of 
motion of a molecule become equally probable after a 
collision with an ion, it was shown that formula (2) reduces to 

U=Xel/m'u, (3) 

as in this case it may be seen that 

l + X m 



m 



(4) 



It will be observed that formula (2) reduces to (1) when 
A, is zero, that is when m is small compared with m\ so that 
either of these two formulae may be applied to the case of an 
electron moving in a gas. Mayer, however, selects formula 
(3) to find the velocity of ions of small mass or electrons, 
although it is definitely stated in my paper that formula (3) 
refers to large ions, and the relation (4) on which it depends 
can only hold when m is greater than m! '. As the correct 
formula (1; for electrons differs by the factor m'/m from 
formula (3), it is unreasonable to expect the latter formula to 
give the velocity of an electron. 

The above formulae, obtained by simple considerations 
when the velocities of agitation of all the ions are taken as 
being the same, are of course not absolutely exact. There 
is a numerical factor by which the expressions should be 
multiplied in order to allow for the variations of the velocity 
of agitation about the mean velocity. In the most interesting 
case, which is that of electrons moving in a uniform electric 
field, the value of the numerical factor is about '9, but it has 
not been determined exactly. The determination of this 
factor is very difficult, as the distribution of the velocities of 
agitation of the electrons depends on the energy of an 
electron which is lost in a collision, and experiments show 
that the proportion of the total energy of an electron which 
is thus lost depends on the velocity. This problem has been 
fully considered by F. B. Pidduck (Proceedings of the 
London Mathematical Society, ser. 2, vol. xv. pt. 2, 1915), 
who shows that under certain conditions the proportion of 
the velocities which differ largely from the mean velocity 
of agitation is much less than the proportion indicated by 
Maxwell's formula for the distribution. 

It appears that the error introduced by taking the velocities 
of agitation as being all equal to the mean velocity may be no 
greater than when the velocity distribution is taken as being 
the same as that given by Maxwell's formula. 

[n order to obtain an exact formula for the velocity U it 
Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. 2 



386 Prof. H. A. McTaggart on the Electrification 

would be necessary to take into consideration the variation 
of the mean free path of an electron with its velocity of 
agitation, and the large reduction of the energy of an electron 
when ionization by collision takes place. 

These points in connexion with the motion of electrons in 
gases have not been taken into consideration by Lenard, and 
it does not appear that his formula is more correct than 
others which have been proposed. 

Yours faithfully, 

3rd May, 1922. JOHN S. ToWNSEND. 



XXXVIII. On the Electrification at the Boundary between a 
Liquid and a Gas. By Professor H. A. McTaggart, M.A., 

University of Toronto *. 

MANY years ago, in the course of some experiments on 
the effect of an electric current on the motion of small 
particles in a liquid, Quincke {Ann. d. Phys. cxiii. p. 513, 
1861) observed that small gas-bubbles in water moved as 
though negatively charged. Although a good deal of atten- 
tion has been paid to the movement of solid and of liquid 
particles in such cases, very little effort has been devoted to 
the study of small spheres of gas suspended in a liquid — one 
obvious reason being the difficulty of controlling them while 
under observation. A systematic examination of their elec- 
trical properties ought, however, to yield further information 
as to the physics — and chemistry too — of surface layers. 

Before the war experiments in this field were begun by 
the author in the Cavendish Laboratory under Sir J. J. 
Thomson, and some results were obtained. Measurements 
were made ^Phil. Mag. Feb. 1914, p. 297) of the velocity, 
under a fall of potential, of small spheres of air in distilled 
water and their electrical charges were estimated. The 
effects on the charge of the addition of minute amounts of 
various inorganic electrolytes were studied. Results were 
obtained (Phil. Mag. Sept. 1914, p. 367) showing how the 
charge varies with the presence in the water of certain 
alcohols and organic acids, and a parallel was shown to exist 
between the variation of the electric charge and the surface 
tension. 

The present paper deals with some further experiments 
carried out in the University of Toronto, and describes the 
variation observed in the electric charge on small spheres 

* Communicated by Professor J. C. McLennan, F.R.S. 



at the Boundary between a Liquid and a Gas. 387 

of air when a particular electrolyte, Thorium Nitrate 
[Th(N0 3 )J, was dissolved in water. This salt was selected 
tor special study because it had been found to be unusually 
active in charging these surface layers. 

The apparatus used was similar to that referred to in a 
former paper, one or two changes being made in it for 
greater convenience. The arrangement is shown in fig. 1. 



Fig. 1. 

K 



- T BT 



J OlD 




A is a small cylindrical glass cell rotating about its axis 
on pivots and driven by a belt of thread from a pulley F on 
a Rayleigh motor. This motor was made in the laboratory 
workshop, and has, instead of the usual fly-wheel with a 
hollow rim filled with water, a solid brass wheel H — a modi- 
fication suggested by Professor Wilberforce of Liverpool. 
The wheel, although loose on the shaft, has enough friction, 
when a heavy oil is used for lubricant, to keep the shaft in 
steady motion after synchronism with the tuning-fork is 
attained. 

D is a timing device consisting of a vertical post carrying 
a pointer and made to rotate by a toothed wheel working in 

9 Q v 



388 Prof. H. A. McTaggart on the Electrification 

the worm E. The pointer rests by its own weight on the 
top of the post, but at any instant in its motion over the fixed 
dial D it may be raised and stopped by a small electromagnet 
controlled by the key B. When released it falls back on the 
post and begins to record time with the same regularity as 
the tuning-fork. It forms a very convenient stop-watch if 
velocities are to be measured. 

A travelling microscope M measures the distance travelled 
by any bubble on the axis of the rotating cell. 

The water used was twice distilled — the second time in 
" Pyrex " glass and condensed in a silver coil. 

The thorium nitrate was by Merck, and was assumed to 
have 12 H 2 — water of crystallization. 

A stock solution was made up containing 4 x 10~ 6 equiva- 
lents per c.c. (1/250 normal), and from this other solutions 
were made by successive dilation. 

A first series of readings was taken with various concen- 
trations of the salt, but with bubbles of nearly the same size 
in order to reproduce the effects previously observed — the 
method of working being to fill the cell A with the desired 
solution, introduce a single bubble of air with the gas pipette, 
and set the cell in rotation. The bubble very soon takes up 
a steady position on the axis, and its motion under any fall 
of potential L may be examined. 

Very small concentrations sufficed to reduce to zero the 
natural negative charge found in pure water and to give 
the small sphere of air a positive charge. 

The following readings are typical : — 

Fall of potential 34 volts per cm. 

Diameter of bubble 0*3 mm. 

Concentration. ^. f Velocity of 

Equivalents ^ n ot bubble. 

per c.c. c lar S e - cms./sec/volt./cm. 

4X10~ 7 + 5xl0~ 4 

4X10-8 m . „ + slower 

8X10" 9 + very slow 

(1-5X10- 4 ) 

5-7X10'" 9 - slow 

47X10- 9 - faster 

4xl0 -9 - faster 

Pure water — 4xl0 -4 



at the Boundary between a Liquid and a Gas. i^89 

The zero point was reached at a concentration o£ about 
7 x 10~ 8 , a result rather higher than that given in a former 
[>aper. The salt was an entirely different sample, and may 
not have contained the same proportion of water of crystal- 
lization. (See Abegg and Auerbach, ' Inorganic Chemistry.') 

A series of readings was then taken for spheres of air of 
different sizes, one object being to observe the charge on 
very small spheres. It is very difficult, by the use of any 
kind of pipette, to introduce into the rotating cell bubbles 
smaller than 1/5 mm. in diameter. To avoid this difficulty 
the following mode of working was adopted. The solution 
was first placed in a partial vacuum to remove as much 
dissolved air as possible, and afterwards poured into the cell. 
A bubble into this gas-free solution slowly decreased in size 
by absorption until it vanished, while the electric charge 
could be observed at any stage. 

Under these circumstances it was found that for a suitable 
concentration of solution a sphere of air which began with a 
small negative charge almost invariably and in a regular 
way reduced its charge to zero, and gradually took on a 
positive charge. 

The following readings illustrate this point — 

No. 1. 



Concentration. 

Equivalents 

per c.c. 

~ 9 Xo-7 


Diameter of 
sphere 
in mm. 

0-26 

0-17 


Sign 
chare 




- 




0-14 


- 




0-10 







0-08 


+ 



No. 2, 



10- 



Concentration. 

Equivalents 

per c.c. 

X5*7 



Diameter of 
sphere 
in mm. 

0-44 

0-35 

0-26 

0-17 

0-14 



Sign of 
charge. 



390 Prof. H. A. McTaggart on the Electrification 

No. 3. 



Sign of 
charge. 



Concentration. 

Equivalents 

per c.c. 

i~ 9 X5-7 


Diameter of 
sphere 
in mm. 

0-62 




0-53 




0-39 




0-17 




0-08 



No. 4. 



Concentration. 

Equivalents 

per c.c. 


Diameter of 
sphere 
in mm. 


Sign 
charg 


»~ 9 X5-7 


0-71 







0-53 


— 




0-44 


— 




0-35 







0-32 


+ 




0-23 


+ 



Concentration. 

Equivalents 

per c.c. 

10- 9 x5-7 



No. 5. 



Diameter of 
sphere 
in mm. 

0-28 

0-17 

O08 

005 



Sign of 
charge. 



It will be seen from the first four examples given that at 
a concentration of 10~ 9 x 5*7 the change of sign occurs in 
every case. Rarely, as in No. 5, and then only when the 
original sphere was small, did the sign remain the same. 
Even then the charge grew steadily less. In practically 
every case the negative charge slowly decreases as the 
bubble gets smaller, passes through zero, and increases to a 
small positive value. 



at the Boundary between a Liquid and a Gas. 39 1 

Three examples are given for slightly greater concentra- 
tions : — 



No. 6. 



Concentration. 

Equivalents 

per c.c. 


Diameter of 
sphere 
in mm. 


Sign 
charo 


10- 9 x6-6 


0-53 







041 







0-35 


+ 




0-17 


+ 




0-08 


+ 



No. 7. 



Concentration. 

Equivalents 

per c.c. 

10" 9 X6-6 



Diameter of 
sphere 
in mm. 

0-35 

0-28 
0-26 
0-17 
0-14 



Sign of 
charge. 



Concentration. 

Equivalents 

per c.c. 

10" 9 x8 



No. 8. 

Diameter of 
sphere 
in mm. 

0-44 

035 

017 



Sign of 
charge. 

Almost zero. 



Above a concentration of 10~ 9 X 8 the bubbles were always 
positive. 

The examples given show that the spheres do not all have 
the same size when they reach the zero — isoelectric — point 
in a given solution. The larger a sphere is at the beginning 
the larger it is when its charge becomes zero. This suggests, 
as the cause of the change in sign, a kind of coagulation of 
something in the free surface. 

It is known that, in a solution of thorium nitrate in water, 



392 Prof. H. A. McTaggart on the Electrification 

hydrolysis occurs with the formation of thorium hydroxide 
thus — 

Th(N0 3 ) 4 + 4 HOH— >Th(OH) 4 + 4 HN0 3 . 

There is present in the solution some of the original salt, 
some acid, and the hj-droxide in colloidal form. The pre- 
sence of the last-mentioned was suspected as one of the 
causes producing the reversal of sign, and experiments were 
then made to test its activity in altering the charge. 

A colloidal solution of thorium hydroxide as free as 
possible from salt and acid was prepared by dialysis (Burton, 
4 Physical Properties of Colloidal Solutions,' 2nd Ed., p. 16). 
A dialysing " sleeve " shaped in the form of a test-tube was 
made of " parlodion " (sold by the Du Pont Chemical Co., 
New York). A solution of the parlodion in ether and 
alcohol was used to coat the inside of a test-tube of suitable 
size. After the solvent had evaporated the parlodion re- 
mained as a thin but strong film which when detached from 
the glass served very well as a dialysing vessel. 

For this experiment a solution containing about 2 gm. of 
salt in 50 c.c. of water was dialysed for a period of three 
weeks, after which an estimate was made of the colloid pre- 
sent. A sample of 10 c.c. evaporated over sulphuric acid 
gave a residue of '0034 gm. The residue formed a thin 
layer of gelatinous material on the bottom of the evaporating 
dish, with drying cracks across it in all directions. 

The effect of this colloid on the charge on small spheres 
of air in water was then examined, the dialysed solution 
above mentioned being diluted as shown in the following 
examples : — 



No. 


C.c. colloid 

solution in 

100 c c. water. 


Diameter of 
sphere 
in mm. 


Sign 
char 


1 


10 


0-21 


+ 






0-12 


+ 






0-07 


4- 


2 


5 


017 


+ 
+ 


3 


25 


0-17 






0-07 


+ 



It is seen that the surface is charged positively by the 
presence of very small amounts of the colloid. 

The following examples show the gradual reversal of the 



at the Boundary between a Liquid and a Gas. 393 

sign of the charge accompanying the absorption of the 
bubble : — 



No. 



C.e. colloidal 

solution in 

100 c.c. water. 


Diameter of 

sphere 
in nun. 


Sign of 
charge. 


1-0 


0-17 


— 




0-14 


+ 


0-5 


0-35 


_ 




0-26 


— 




012 


+ 




005 


+ 


0-25 


032 


- 




0-17 


- 




0-08 


— 




0-05 


4- 



The experiments show that the colloidal thorium hydroxide 
gives both the effects observed with the ordinary solution. 
It not only charges the surface positively if present in suffi- 
cient amount, but it also exhibits the reversal of charge with 
diminishing size of the bubble, and this, too, in concentrations 
of thorium of about the same order as in the case of the salt. 

Discussion. 

The state of the matter and the nature of the electric 
forces in surface layers of liquids is still a subject on which 
no very clear ideas exist. Experiments on electro-endosmosis 
all point to a selective action in such layers so far as the 
ions in the solution are concerned. But (he observations 
are always complicated by the presence in contact with the 
liquid surface of a solid whose role in the selecting we are 
ignorant of. The same is true of cataphoresis experiments 
with solids, as, for example, in the study of the electrical 
charge on colloidal particles. This difficulty is avoided, 
however, in similar experiments with small spheres of air — 
or any gas — and in such cases we can safely regard any 
effects observed as due largely to the properties of the liquid 
and its free surface. In particular, the electrical charge 
existing at any air-liquid surface may be considered as the 
result of forces residing altogether in the liquid. It ought 
to be possible, then, in considering potential differences at 
solid-liquid junctions to isolate the contribution of the liquid. 

In the case of thorium nitrate in solution the selective 



394 Electrification at Boundary between Liquid and Gas. 

action of the air-water surface is very marked, a positive 
charge being acquired by the surface with very minute 
concentrations of the salt. The positive ions available for 
selection are Th + and H + , but neither of these separately 
can be responsible for the unusual activity of the salt. The 
mere presence of H + ions, as, for example, in the form of an 
acid, does not produce so great an influence on the surface 
charge. Nor can free Th + ions have much effect, for they 
disappear in the dialysis and yet leave the pure colloidal 
solution practically as active as before. The real agent 
must be the particles of colloidal thorium hydroxide which 
gather about them groups of H + ions and carry them into 
the surface in larger numbers than would be possible for the 
H + ions alone. 

The nature of this selective action must be connected with 
the shape of the surface, or, to put it in another way, a 
particle must reach a certain size before it can be regarded 
as having a surface-layer about it with a tension and an 
electric charge. We have at present in order of size — ions, 
ionic micelles (Prof. McBain, " Soap Solutions," Nature, 
March 10, 1921), ultra-microscopic colloidal particles, micro- 
scopic and macroscopic particles including gas-bubbles. At 
what stage a surface-layer is formed it is difficult to say, but 
it seems reasonable to suppose that the curvature of such a 
surface would have an effect on the charge adsorbed. The 
change of sign with decreasing size of air-sphere shown in 
these experiments seems to bear out this idea. 

The information obtained regarding the effect of thorium 
nitrate on the electrification of air-water surface layers may 
be summarized as follows : — 

1. Thorium nitrate in aqueous solution and in concentra- 
tions as small as 8xl0~ 6 normal gives a positive electric 
charge to the surface of a sphere of air immersed in it. (In 
distilled water the charge is always negative.) 

2. For concentrations in the neighbourhood of 6 x 10~ b 
normal a sphere initially negative becomes gradually positive 
as the sphere diminishes in size. 

3. Colloidal thorium hydroxide in small concentrations of 
the same order also gives a positive electric charge to a 
sphere of air immersed in it. 

4. Colloidal thorium hydroxide also exhibits the reversal 
of the sign of the charge with a decrease in the size of the 
bubble. 

5. It is suggested that this reversal of sign is experimental 
evidence of a relation between the curvature of the surface 
and its adsorptive power. 



Lecture-Room Demonstration of Atomic Models. 395 

The experiments are being- continued as time permits in 
the hope of obtaining* some new information regarding these 
free surfaces. Is is the intention to compare with thorium 
the effects of one or two other tetravalent and trivalent 
metals in the colloidal state. 

I wish to thank Professor J. C. McLennan for his kind 
and encouraoino- interest in the work. 



XXXIX. Note on a Lecture-Room Demonstration of Atomic 
Models. By Louis V. King, D.Sc, Macdonald Professor 
of Physics, McGill University*. 

[Plate II.] 
Section 1. 

SEVERAL mechanical models illustrating various types 
of atomic structure have been proposed from time to 
time. Among these we may mention Mayer's classical 
experiments with floating and suspended magnets, illus- 
trating the action of atomic forces t. 

Many modifications of these classical experiments have 
been suggested. In particular, a paper by R. Ramsey de- 
scribes interesting modifications of the original apparatus J. 

Actual apparatus illustrating the supposed structure of 
atoms can now be obtained ready for use from scientific 
instrument makers §. 

All these methods involve the repulsive forces between 
steel elements (needles or spheres) in a permanent magnetic 
field, together with the central attraction set up by a per- 
manent magnet. An important point contributing to the 
success of the experiment is that all the magnets, repre- 
senting electrons, have as nearly as possible equal pole 
strengths. Owing to magnetic reluctance and effects of 
demagnetization, these conditions are difficult to realize in 
practice without a considerable amount of care and ex- 
penditure of time. 

* Communicated by the Author. 

t J. J. Thomson, ' Corpuscular Theory of Matter * (1907), Chapter 6, 
pages 103 et seq. 

X It. R. Ramsey, "The Kinetic Theory of the Electron Atom." Pro- 
ceedings of the Indian Academy of Sciences, 1918. Phil. Mag - , vol. xxxiii. 
Feb. 1917, pp. 207-211. 

§ W. M. Welch, Scientific Company, Chicago. 



396 Prof. L. V. King on a Lecture-Room 

Section 2. 

The magnetic elements which form the essential feature 
of the apparatus to be described consist of a number of steel 
spheres or small soft-iron rods magnetized in a strong 
alternating field. 

One such model is shown diagrammatically in PL II. fig. 1, 
while fig. 2 shows the actual apparatus. The coil A consists 
of 340 turns of number 12 B. k S. copper wire (2 mm. diam.); 
inside radius of winding 8"8 cm., outside radius 13'5 cm., 
width of coil 3" 9 cm. Such a coil has a resistance of 
approximately 1*3 ohms and self-inductance of about 32 
millihenries. It may be connected directly to a 110-volt 
60-cycle A.O. circuit without overheating. In such cir- 
cumstances it draws a current of about 9 amperes. It is 
approximately of such dimensions as to give a maximum 
field strength at the centre of the coil. 

Placed over the opening of the coil is a large watch-glass 
B whose radius of curvature is approximately 25 cm. If 
available, an accurately ground concave glass mirror may 
be used to advantage. If, now, a supply of steel ball-bearings 
about 3 mm. in diameter is available, these may be placed 
on the concave surface B, where they will experience an 
attraction towards the lowest point approximately pro- 
portional to the distance. When the maximum current is 
passed through the magnetizing coil, the steel spheres will 
become A.C. magnetic doublets of very uniform magnetic 
moments. It will be noticed that the magnetic axis will 
always be very accurately along the direction of the mag- 
netic field, independently of the rolling motion of the balls. 
Furthermore, if the spheres are of fairly uniform quality and 
the field strength sufficiently great, the instantaneous mag- 
netic moments of these doublets will be equal in magnitude 
and phase. In these circumstances the steel spheres will 
repel each other with a force varying as the inverse fourth 
power of the distance, the constant of proportionality being 
accurately the same for all the spheres. With the attraction 
to the centre varying as the distance, it maybe expected that 
the magnetic elements will form remarkably symmetrical 
stable groupings. One such grouping is illustrated in 
PI. II. fig. 3 (a). 

It is obvious that by a very simple arrangement of lenses 
and mirrors this model atom may be projected on a screen. 
The concave surface B may, if desired, be mounted so as to 
allow of rotation, thus increasing the interest of the " atomic " 
arrangements. This experiment is extremely convenient for 
lecture-room purposes, as it requires no preparation and is 



Demonstration of Atomic Models. 391 

always certain to give results which never fail to delight an 
audience. 

An interesting variant of this experiment is to make use of 
the arrangement of two coils described in Section 5 (figs. 
5 & 6). A surface of clean mercury is placed midway between 
the two coils. A number of steel balls floating on this surface 
will repel each other as already described, and will all tend 
towards the centre, owing to the greater intensity of field. 
The remarkably regular arrangement taken up under these 
conditions is shown in fig. 3 (6). The damping is so slight 
that the system may be set into oscillation in various ways 
by means of external magnets, giving a good illustration of 
internal vibrations in the atom. It would, moreover, be 
possible with no very great expenditure of labour to deter- 
mine the frequency of various modes and compare the results 
with theoretical calculations. 

Section 3. 

The same apparatus may also be used to illustrate the 
motion of the molecules of gas or the Brownian movements. 
For this purpose an elongated piece of iron is employed, 
e. g. a short cylinder of iron or steel wire about 1 cm. in 
length by 1 mm. in diameter. In the alternating field of the 
coil such a magnet experiences a very strong torque, which 
vanishes when the axis lies along the direction ot the 
resultant A.O. field. If such a magnet is placed in a flat 
cylindrical glass vessel occupying the centre of the coil, ai.d 
the field suddenly applied, violent movements of the little 
iron rod will be observed. The instantaneous moments set 
up by the field will be sufficient to make the rod leave the 
surface on which it is resting and describe a trajectory 
under the combined effect of gravity and the magnetic field. 
At the termination of the flight, it will again strike the glass 
plate and will then receive an additional impulse made up of 
the magnetic torque and the elastic reaction at contact with 
the glass. This will start it on a new trajectory, and the 
process will be continued indefinitely until the rod makes 
contact with the plate at the termination of its flight in such 
a way that the instantaneous torque is zero. Then it stops 
dead with the axis pointing along the direction of the field. 
This is an event which happens very rarely. Several such 
rods enclosed within a glass vessel will keep in constant 
motion in a manner resembling the motion of molecules in a 
rarefied <:as. An interesting variant of this experiment is to 
insert .-hort steel wires along the diameters of small pith balls 
which hop around, describing flights in the glass vessel as if 



398 Prof. L. V. King on a Lecture-Room 

they were animated with life. As before, the glass vessel 
and its contents may be projected on a screen, the resulting 
effect being illustrative of molecular movements. 

Section 4. — Experiments on Electrodynamic Repulsion. 

Owing to the distribution of the magnetic field around the 
coil employed in cbis experiment, the same apparatus is well 
suited to the demonstration of electrodynamic repulsion. For 
this purpose several plates of aluminium or copper should be 
cut with a radius approximately equal to the outer radius 
of the coil. Such a disk may be anchored by three strings 
fastened at equidistant points of the circumference so as 
to allow it to move vertically, with its centre over the 
axis of the coil, which is laid in a horizontal position. 
On applying A.C. circuit, the plate will float three or 
four centimetres above the coil. By placing a light iron 
rod (3 cm. x 1 mm.^ on the plate, the direction of the 
A.C. field is easily demonstrated, as shown in PI. II. fig. 4. 
It will be noticed that over an annular region bounded by 
the outer edge of the plate and a circle of half its radius, the 
lines of force are inclined at approximately 45° to the vertical. 
It is the reaction of the horizontal component of the A.C. 
field with the induced current due to the vertical component 
which causes the repulsion referred to. To demonstrate this, 
a circular plate may be cut up into several concentric rings 
and laid on a sheet of glass. When current is applied it is 
only the outer rings which are repelled, the force on the 
inner rings gradually becoming less, until that on the central 
disk in a practically uniform field perpendicular to its plane 
is practically nil. 

Iron filings poured on a glass plate laid horizontally over 
the coil assume an interesting laminar distribution, which 
again may be projected on a screen. The iron filings tend 
to arrange themselves in a series of vertical planes about 
1 cm. high arranged radially. It is easily seen that this 
arrangement is due to the fact that under the influence of the 
alternating field, each of the radial planes represents a series 
of vertical A.C. magnets which repel each other. Their 
height is limited by the vertical stability of the plates under 
the combined effect of gravity and of the alternating field. 

Section 5. — Experimental Model of the Rutherford Atom. 

By using two coils of the dimensions already described, 
arranged with their planes horizontal at a distance apart 
equal to the mean radius (Helmholtz arrangement), it is 






Demonstration of Atomic Models. 399 

possible to secure a fairly uniform field over a considerable 
area midway between the coils. Such arrangement (PI. II. 
figs. 5 & G) allows of interesting experiments on a model atom 
approximating more closely to modern ideas. A shallow 
circular basin of mercury is placed on an adjustable stand 
between the two coils. A number of steel pins with glass 
heads serve as the elements (electrons) for the model. It 
oue of these is placed with the glass head on the mercury 
surface, it will float in a vertical position and tend to move 
towards the centre ot the field, owing to the greater concen- 
tration of lines of force. This force towards the centre may 
be varied at will by adjusting the height of the mercury 
surface, or by placing rods of soft iron along the axis of the 
coils at adjustable distances above or below the mercury 
surface. If a second pin be floated on the mercury surface, 
it will repel the first with a force varying nearly as the 
inverse square law when the distance apart is not too great. 
A third pin may be added, when a triangular arrangment will 
be formed. Successive pins give the familiar series of regular 
polygons arranged in concentric rings. It is evident that 
the great advantage of the A.C. field is to make the mag- 
netical polarity of each of the pins very nearly equal, thus 
giving rise to a remarkable symmetry in the arrangements 
formed, as illustrated by figs. 7 (a) and 7(b) (PI. II.). 
As before, the experiment can be carried out in such a way 
that the various stable arrangements may be projected on a 
screen. It is extremely simple to demonstrate the apparatus 
at a moment's notice, the only precaution necessary being to 
use clean mercury so as to allow a great mobility of the 
floating pins on an uncontaminated surface. 

It is interesting to notice that rotation of the basin con- 
taining mercury does not disturb any particular stable 
arrangement, owing to the fact that the centrifugal force 
is accurately balanced by the change of slope of the para- 
boloidal mercury surface. 

The use of an A.C. field allows of the possibility of realizing 
positive electrons and a central nucleus, the law of forces 
between them being very nearly that of the inverse square 
and at the same time very exactly that corresponding to 
charges of ±e, ±2e, ±3^, etc. It is evident from fig. 5 
(PI. II.), illustrating the model under consideration, that 
electrons may be represented by lengths of soft-iron wire of 
the same diameter arranged to move with both ends in the 
same plane at distances not too far apart compared with their 
length. In these circumstances we have repulsion according 
to the inverse square law, the charge — e being represented 



400 Lecture-Room Demonstration of Atomic Models. 

by the average pole strength ±?n of each rod, which is 
extremely uniform. A nucleus of positive charge ne may be 
made up by taking 2n lengths of the same wire and inserting 
them in a small glass or aluminium tube, as shown in fig. 5, 
illustrating a nucleus of charge + 2e. In these circumstances, 
each of the rods representing electrons is attracted to the 
nucleus with a force varying nearly as the inverse square of 
the distance and proportional to nm x m } the average pole 
strength of each end of the rod being ztm. 

In order to realize this arrangement, the rods (about 
7 cm. x 1 mm. diameter), representing negative electrons, 
should be suspended from silk fibres about 1 metre or more 
in length. By adjusting the position of the rods in the 
space between the coils, a position of neutral equilibrium 
may be found in which there is practically no tendency for 
the rods to move either towards the centre or radially 
outwards. Under the combined effect of gravity and of 
the magnetic field they seem to float in any position. When 
this adjustment has been made, the rods representing the 
nucleus should be set in position along the axis of the coils. 
The suspended rod representing the electron may then be 
projected so as to describe a path about the fixed nucleus, 
and a damped elliptic orbit will be observed, the nucleus 
being at one focus. . 

If two lengths of wire are used to make up a nucleus -\-e 
in the manner illustrated by fig. 5 (a), we obtain a model of 
the hydrogen atom which is dynamically stable. 

If we make up positive nucleus of charge 2e, represented 
by two pairs of iron rods, we obtain a model (fig. 5 (b)) 
of the ionized helium atom which is dynamically stable. 
If we introduce an additional iron rod representing an 
electron (fig. 5), and therefore a complete helium atom, it 
seems impossible to obtain a dynamically stable arrangement 
by any circumstances of projection. For instance, any 
attempt to reproduce the symmetrical oscillation suggested 
by Langmuir meets with failure, owing to the dynamical 
instability of this arrangement. 

It is obvious that further experiments along these lines, 
leading possibly to results of great interest, might be carried 
out by constructing large solenoidal coils to give a uniform 
A.C. field, in which circumstances the inverse square law of 
attraction and repulsion between electrons and nuclear 
charges ne (n=l, 2, 3, etc.) would be faithfully reproduced. 



L 401 ] 



XL. The Influence of the Size of Colloid Particles upon the 
Adsorption of Electrolytes, By Humphrey D. Murray, 

Exhibitioner of Christ Church, O.vford*. 

^EVERAL workers have examined the influence of con- 
k^ centration upon the coagulation of colloidal solutions, 
hut references to the effect produced by alteration in the 
degree of dispersion are few and not very definite. Kruyt 
and Spek t examined the coagulation of colloidal arsenious 
sulphide, and found that the coagulative value of univalent 
ions increased with increasing dilution ; in the case of a 
divalent ion there was a slight decrease ; whilst for a ter- 
valent ion there was a rapid decrease in the coagulative 
value. Burton and Bishop + examined the coagulative 
values of various ions upon colloidal solutions of arsenious 
sulphide, copper, and gum mastic, and as the result of their 
experiments found that with univalent ions the concentration 
of the ion required for coagulation increased with decreasing 
concentration of the colloid, for divalent ions the concentra- 
tion of the ion was nearly constant, for trivalent ions the 
concentration of the ion varied almost directly with that of 
the colloid. More recently YVeiser and Nicholas § have 
extended these researches to colloidal solutions of hydrous 
chromic oxide, prussian blue, hydrous ferric oxide, and 
arsenious sulphide. They found in the case of the first three 
that the coagulative values of electrolytes tended to 
increase with dilution of the colloid, but the increase was 
less marked with electrolytes having univalent precipitating 
ions, and became more marked as the valency rose. Oden 
found that sols with ultramicroscopic particles are more 
sen-itive to electrolytes than those containing amicrons. 

The object of these experiments was to examine the 
influence of the size and uniformity of colloid particles upon 
the adsorption of electrolytes as measured by the minimal 
concentration for coagulation. For this it was necessary to 
obtain solutions of the same colloid prepared under identical 
conditions, but containing particles of different mean size. 
It was decided to employ Oden's method of fractional 
coagulation. The most suitable colloid to use, therefore, is 
one which, when first made, contains particles of markedly 

* Communicated by the Author. 

t Kruyt and Spek,*AV/. Zeit. xxv. p. 1 (1919). 

X Burton and Bishop, Jour. Phys. Chern. xxiv. p. 703 (1920). 

§ Weiser and Nicholas, Jour. Phys. Chem. xxv. 742 (1921). 

Phil. Mag. S. 6. Vol. 44. No. 260. Aug. 1922. 2 D 



402 Mr. H. D. Murray on Influence of Size of Colloid 

different size, and is stable when precipitated, redispersed, 
and dialysed. Gum mastic was found best to meet the 
requirements, and was used in the subsequent experiments. 
To show that the solutions employed were comparatively 
stable, the concentration of NaCl required to precipitate one 
of the fractions at the beginning and end of the experiments 
was measured and found to be : — 

Feb. 15th 433 millemols. 

Mar. 29th 439 



Fractionation. 

One gram of finely-powdered picked gum mastic was 
dissolved in about 20 c.c. of alcohol, and poured slowly with 
vigorous stirring into one litre of distilled water. By this 
method seven litres of mastic solution were prepared. Oden 
recommends that in all cases NaCl should be used for the 
precipitation. With mastic this necessitates a very large 
concentration of salt, which appears to be strongly adsorbed, 
and comes slowly through the dialyser. It was thought 
better to employ HC1, which precipitates in smaller concen- 
tration. It was found convenient to separate the mastic into 
seven fractions with these concentrations of HC1 : — 

Concentration of HC1 Condition of 
Fraction, in millemols. Precipitate. 

I 0-1-1 trace 

II ri-l"4 good 

III 1-4-1-7 

IV 17-2-0 

V 20-2-3 

VI 2-3-2-6 small 

VII. 26-2-9 trace 

The procedure was as follows: — 200 c.c. of the mastic 

N 
solution were mixed with a quantity of ^0^01 *° §^ Ye * ne 

required concentration, and then poured into the centrifuge 
vessels and allowed to stand for 60 minutes from the moment 
of mixing. It was then centrifuged at 3000 r.p.m. for 
30 minutes. At the end of this time the supernatant liquid 
was poured off and the precipitate carefully shaken up with 
about 100 c.c. of distilled water. Fractions II. and VI. were 
retained until about 1500 c.c. of each had accumulated ; the 
other fractions were rejected. At the same time 1500 c.c. 



Particles upon the Adsorption of Electrolytes. 403 

of mastic were completely coagulated with a concentration of 
3*0 millemols. of HC1, and redispersed in an equal quantity 
of water. It appears below as solution B. It is to be 
expected that Fr. II. will contain particles of an average size 
greater than those in Fr. VI. and both will contain particles 
of more uniform size than those in solution B. 

The solutions after dispersion were kept in dialysers of 
parchment paper until the dialysate was uncontaminated 
with HC1. They were then placed in perfectly clean vessels 
of resistance glass fitted with a siphon, and a soda-lime tube 
attached to the air-inlet. The siphon pipes were closed by 
short pieces of rubber tubing and pinch cocks. 

Basis of Comparison. 

Any method of comparison between two or more solutions 
based upon the total masses of the disperse phase in unit 
volume is useless when applied to data due to adsorption. It 
is possible to take as a basis the number of particles in unit 
volume, or, what is probably more characteristic and capable 
of giving more directly comparable results, the total inter- 
facial surface in unit volume. The former may in most cases 
be ascertained by a direct count under the ultramicroscope. 
To evaluate the latter it is necessary, beyond this, to know 
the total mass of the disperse phase, which can be effected by 
weighing after evaporation to dryness, or by the methods of 
volumetric analysis. In addition it demands a knowledge of 
the density of the disperse phase, or of the specific gravity 
of the solution and of the dispersion medium. 

Perrin in his researches upon Brownian movement 
obtained the density of the mastic with which he was working 
by evaporating- a portion of his suspension to dryness and 
estimating the density of the solid mastic. This value 
(1'064) he found to agree admirably with the density as 
determined from specific gravity measurements. It seems 
uncertain, however, as Burton * has pointed out, whether it 
is justifiable to assume that the density of the particles in the 
ordinary colloidal solution of gum mastic is the same as that 
of the solid substances. Perrin, as a matter of fact, used a 
suspension of mastic which had been obtained by centrifnging 
the larger particles from a solution of mastic and rejecting 
the remainder. In the case of the present solution, it 
seemed desirable to determine the density of the particles 
directly, with a pvknometer. 

* Burton, ' Physical Properties of Colloidal Solutions,' 2nd Edition, 
p. 125. 

1 D 2 



404 Mr. H. D. Murray on Influence of Size of Colloid 
Concentration of Mastic. 

Thirty c.c. of the three solutions were evaporated slowly 
to dryness in a steam oven, and, as a mean of several deter- 
minations, gave the following weights of mastic in 10 c.c. of 
solution : — 

Weight found. 

Solution B -00463 gms. 

Fr. VI -00171 „ 

Fr. II -00339 „ 

Number of Particles. 

A true ultramicroscope was not used to count the particles, 
but a cardioid condenser, fitted to an ordinary microscope. 
The chief difficulty in work of this nature is to ascertain 
accurately the volume of the liquid within the field of view. 
A cell was made according to the recommendations of 
Siedentoff'*, the only alteration made being the substitution 
of heavy glass for fused quartz. Fluorescence due to the 
glass was not sufficient to render difficult the counting of the 
comparatively large mastic particles. The cell consists of a 
glass plate, 5 cm. in diameter and 1*0 mm. in thickness, 
provided with a circular groove. The portion enclosed by 
the groove, 1 cm. in diameter, was polished exactly 2 /jl 
deeper than the surface of the plate. This was used with a 
cover slip about *25 mm. in thickness. The cell was soaked 
in concentrated sulphuric and chromic acids, washed with 
water, and then passed through two solutions of re-distilled 
alcohol. It was finally flamed. The source of illumination 
was a Pointolight lamp, fitted with a condenser. All the 
solutions examined were diluted with water which had been 
carefully distilled and allowed to stand for a month undis- 
turbed. It contained on an average 1 particle in 20 
counts in a volume of 14'1 x 10" 5 cu. mm. and could, there- 
fore, be considered optically pure to the degree of accuracy 
to which work was carried. All the solutions were contained 
in vessels of resistance glass, closed with corks covered with 
tinfoil. The method of procedure was to transfer, by 
means of a clean platinum loop, a very small drop of the 
solution to be examined to the central portion of the cell. 
The cover slip was laid on and pressed down until the 
Newton interference rings appeared at the edges. The 
dilutions were such that, when viewed with a convenient 
stop in the eyepiece, about three or four particles appeared 

* Siedentoff, Verhd. Deut. Phys. Ges. xii. p. 6 (1910). 



Particles upon the Adsorption of Electrolytes. 405 

in the field of view. One hundred counts were taken at half- 
minute intervals, and the average number deduced from this. 
A few of the {(articles, especially in the case of Fr. II., 
tended to adhere to the walls of the cell, and to prevent 
any error due to this, the field of view was shifted five 
times during- each count. 

The results obtained were as follows : — 



Solution. Dilution. 



Soln.B.. x72G 

Fr. II. . . x 396 

Fr.VL.J X396 

I 



Ob- 
ject iA-e. 



'4 mm. f.l. 



Eye- 
piece. 



jDiam. of 

, Field of 

vieAV. 



X 12 -30 mm. 
X 12 -30 „ 
Xl8 I -20 „ 



Volume of Field ^ '% 

of view. t-, ,'. , 

Particles, 



14'1 X 10 - 5 mm. 
14-lxlO- 5 „ 
6-28 X 10 - 5 „ 



4-1 
30 
4-0 



Density of the Particles. 

The density of the solutions was determined with an 
accurate pyknometer in a thermostat at 17*2° C. The 
weighings agreed to *0002 gm. Fr. VI. was too dilute to 
give accurate results. The specific gravity of each solution 
rose slightly during dialysis, owing probably to the removal 
of adsorbed or dissolved alcohol. This rise continued for 
about five days. The weighings were made at the end of 
ten days. The dialysis was then continued in more efficient 
dialysers made by Soxhlet thimbles impregnated with 
collodion, but the specific gravity remained constant. As a 
mean of four weighings for each solution, the following 
values for the density of the particles were obtained : — 

Soln. 3. = 1-195. 
Fr. II. =1-186. 

As a mean the density of the mastic was taken to be 1*190. 
This value is considerably higher than the density of the 
mastic in bulk, owing possibly to changes occurring either 
on dispersion, or coagulation. Perrin * states that he 
observed the density of his carefully washed granules 
apparently to rise in salt solutions, and this may account in 
part for the difference. 

Size of Particles. 

From these three sets of data — the number of particles in 

unit volume, the total mass of mastic in unit volume, and the 

density of the particles— it is possible to calculate the mean 

* Perrin, Ann. C'him. Phys. xviii. p. 5 (1909). 



406 Mr. H. D. Murray on Influence of Size of Colloid 

radius r of the particles in each liquid. 01' these three 
measurements that of the density seemed possibly least 
accurate, but, as it occurs in each calculation, the relative 
sizes remain unchanged. 



Solution. 



Soln.B. 
Fr. II.. 

Fr. VI. 



No. of 
Particles 
in mm. 3 

(=»). 



211x10 s 

84-2 xlO 5 
252 x10 s 



1 

Total Massi 

of Particles 

in 10 c.c. 


•0046 gm. 
•0034 ,. 

•0017 „ 

1 



Mean volume 

of one 

particle. 


1-83X10- 2 

3-40x10-2 

•57x10-2 


J" 3 



Radius 
(=r). 



r 2 X»XlO -3 . 



•164 p 
•201 p 
■111 JU 



564 
340 
307 



r 2 xn is a measure of the interfacial surface in unit 
volume. 

Borjeson * has successfully combined the principle of 
gilding metal particles with observation of the rate of sedi- 
mentation of the particles so gilded, to measure the size of 
the original particles. He failed to obtain successful results 
with gelatine and gum arabic sols. An attempt to apply 
this method to the mastic solutions as a check on the results 
obtained also met with failure. It appears therefore unsuit- 
able for organic colloids. 

Rate of Coagulation. 

It has been customary to fix an arbitrary time during 
which the colloid solution is allowed to stand after the 
addition of the electrolyte and before the amount of 
coagulation is measured. Burton f allowed the solutions 
which he examined to stand ten hours ; and again J the more 
dilute solutions which he examined were left for "some days." 
Weiser and Nicholas § allowed the solutions under examina- 
tion to stand for twenty-four hours. 

In some preliminary experiments the writer found that 
abnormal results were obtained with a dilute solution owing 
to the fact that the time elapsing before examination was too 
short to permit of coagulation with the minimal quantity of 
electrolyte. This led to an examination of the actual rate of 
coagulation. Two solutions were employed, one being ten 
times more dilute than the other. The results w ere as 
follows : — 

* Borjeson, Koll. Zeit. xxvii. p. 18 (1920). 

t Burton and Maclimes, Jour. Phys. Chem. xxv. p. 517 (1921). 
X Burton & Bishop, Jour. Phys. Chem. xxiv. p. 703 (1920). 
§ Weiser and Nicholas, Jour. Phys. Chem. xxv. p. 742 (1921). 



Particles upon the Adsorption of Electrolytes, 407 



2 S 



m 



o 



5 'o 



- 
CR9 



S! 


w2 


c 








° - 


00 tT 








— 2- 


9 




aq £L 




<s 




E9 



SI 


jT 


O 




CI 


-*• 




CO 5 




ft 










US 




S 










0* 


M 








75 





b 



o' 

g 

o" 



3 



x 



O 







t-f (-"O 


g 


'-'Ccai.aj-ai-^cco 




Or005H05rf.OO 


5 




ee 


5 


p 


p; 


H? 


S S S 3 « 2 S aT 


2 




B 
o 


g 


CD 


R 










--.».. ^ j _ S-' 


CTC 






ct> 




u_i Q 




£ ~ 




o s 




O _ i— ■ 




2 § 3.dS" 


Ci 


O 3 rf r3- 




s o s *a s - Ei^ 


Br" 


c 


p " S-" " p'g. 




3 «> -s S: 

«8 ^ •Sgs 


CO 


JL P5QTQ 




CD O 




P O 




4 ^ 




m 




o 








® 




P 




<-s 




SI * 


to 


O £-' 


!£ 


o tr 


cr 


z cr» s - ^ s ^_ 


o 


p » 




3 J2 




1 ^ 


DO 


PS 




O 




o 




*r 




CO 




si 


£ 


rf^ 


O J=3 




s Cr 1 - ; : ; j' » 






P P. 


o 




C 


cfq 

B) 


? 



g 

o" 



o 

O 
CD 

O 

CD 
P 



hj 



02 

CO 



c 



408 Mr. H. D. Murray on Influence of Size of Colloid 

It is obvious that the rate of coagulation decreases con- 
siderably with decreasing concentration of the mastic. The 
point of complete coagulation was taken to be " clear with 
large flocks." These flocks do not necessarily settle to the 
bottom of the vessel ; some adhere to the side. The interior 
of the liquid, however, appears quite clear. It is noticeable 
that the flocks adhering to the sides are more numerous 
upon, if not confined to, the side of the vessel away from 
the source of daylight illumination. 

It is apparent that, in order to arrive at the point of 
complete coagulation in the case of the more dilute solution, 
it is necessary to leave it undisturbed for a good many days, 
a course of action to which there are several objections, 
apart from that of: mere convenience, in carrying out a series 
of numerous determinations. During this time external 
influences, such as chemical action at the surface of the 
particle, have more time to show themselves. 

These objections can be obviated by centrifuging the 
solution after a definite time at a constant speed. The 
rate of coagulation is made up of two factors, the rate of 
aggregation of the particles and that of settling of the aggre- 
gates so formed. By centrifuging, the influence of the 
latter is reduced to a minimum, and we arrive at a truer 
measure of the former. An examination of this method 
shows that it is possible to obtain complete coagulation after 
a reasonable length of time. 

The solutions were treated in the way to be described, and 
the following tables (and figs. 1 and 2) show the minimal 
concentrations of A1 2 (S0 4 ) 3 and NaCl at various intervals 
after the moment of addition for complete coagulation. 
Similar results were obtained with BaCl 2 . 

It will be seen that the minimal concentration of electro- 
l} r te decreases rapidly with time until from 12 to 22 hours 
after mixing, thereafter, it remains fairly constant. In 
order, however, to ensure reaching the true end-point, the 
solutions in the subsequent experiments were allowed to stand 
48 hours after mixing and before centrifuging. 

Several workers have pointed out that the rate and 
method of addition of the electrolyte affect the end-point. 
Weiser and Middleton * devised an apparatus, by the use of 
which they obtained concordant results, and a modification 
of it made by the writer was found to give equally good 
results. In order to ensure perfect cleanliness, it was made 
of glass throughout. The modified apparatus consisted of 

* Weiser and Middleton, Jour. Phys. Chem. xxiv. p. 30 (1920). 



Particles upon the Adsorption of Electrolytes. 



409 



1 
o 




o 




Limina 
ncentra 
of NaC 






g 


o' 

3 


B 






O 


s 




4 


J-; 




P 


5' 


C ' o 


^ 


O | 












l_l 




— o 




c o I 




y 




o 




Q 




V ■" 




t* 4 ,— ' 


» 


Cn Ot 


o 


8 8 








B 


3' 






- cT 


» 

a 




X 










o^ 








JO 




V 




I— 1 




Cn OS 


33 


O CO 


3* 


c o 




— 1 




B 




~ 


3" 


— ! 








B 


n 


o 








to 




cC J^ l 




O Cn 




° ° 


LO 


B 






3 






" l 




c 








GO 




CO rf>. 




•— Or 




c o 


Oi 




r 






S CD 




2 


SI 


c 








OD 




-5 CO 




ro c: 




c o 


tO 


— 


4- 










E3 


3 


- CD 










X 


o_ 




CO 




-J OS 




to o» 




o o 


j^. 




30 














— 




- 2 




B 





£_ 









P 

o 

O 

o 
p 

d 



3C 

- 

0Q 



s 3 



2; 

P 

o 



o P 




»»§ 




^2 t- 1 




,r-2 g- 


3 


OD^f -• 


3* 


*°&£ 


cj"' o 




• 3 


2 


* — v ' 




o 


k 


12 


i 










- ET. an? 


o 




3 




)_i 




i-i o 




o o 




^ 












p 




V V 




tO fcO 




o o 


3l 


3 


3 






f -1 


3 


s cd 


3 


3 




C^ 


c 






CO 




V V 




to to 


Oi 


o o 


© 






B 








;=: 


3 


~ CO 










V 


o 


B 






00 




V 




to 




O HJ 




© 


Di 


? 


— 


B 


3 






ST 


5 

CO 


- 3 




c 








09 




to CO 




CO CO 


_i 


5 


3* 






pi 


3 


| 


CD 






CO 




H^ CO 




en co 


to 


— 


3t 






J2 


3- 



- aT 




o 


£ 






CD 




h^ ci 




Ot 00 


*» 


5 


DO 












3 


- 2 








o 


L 






p 





P 








a> 




o 




| -^ 




'— > 








O 




P 




crq 
















p 




c-f- 








o 


(-/J 





w 


o 


60 


i-rs 


FJ 


P 


GO 
1— i 




HH 






o 




* 





^2 

o 



410 Mr. H. D. Murray on Influence of Size of Colloid 

two vessels, one slightly smaller than the other, and fitting 
by a ground-in joint inside the larger. 

The smaller vessel has a slightly higher inner cylindrical 
vessel, the base of which is concentric with that of the 
outer vessel and fused to it. The electrolyte solution is 
placed in the inner vessel and the colloidal solution in the 
annular space, both having been previously rinsed out with 
their respective solutions. The larger vessel is placed over 
the smaller, and the whole inverted and left for 30 seconds 





















'•J* 






























n e 


i 




£ 


















°aoo 


1 v 

1 Vv 


















v^..^ 






, 10% 








■4 






— ) 












g\g 










i 












.,100% 


,., i 







































Time in hours- after mixing 



W 



to drain. By this means a sudden and complete mixing of 
the two solutions is obtained. The mixed solution is then 
poured into a vessel of hard glass and corked. The whole 
apparatus, as were all the vessels with which the mastic 
solution came in contact, is made of hard glass and was 
steamed out between each series. 

The experiments were conducted with 10 c.c. of the mastic 
solution at the required dilution. Into the inner vessel 
was poured enough water to make the volume of the elec- 
trolyte solution up to 5 c.c, and then the latter solution 
was added at a convenient concentration. To determine 



Particles upon the Adsorption of Electrolytes. 411 

one end point for a given concentration of mastic and a given 
electrolyte, four solutions were made up with a fairly wide 
difference of concentration in each solution, so as to give a 
large bracket. After standing 48 hours and centrifuging 
for half an hoar at 2000 r.p.m., at which speed there was no 
sedimentation of the pure mastic solution, four more 
solutions were mode up, in which the concentrations of 
electrolyte were such as to cover the interval between the 



3-2 



H 
















































Fig 


I 






















































\ 
\ 

\ 

\ 

L \ 
















V \ 
\ \ 

v \ 






,.100% 
















-xJ2&- 






9 : 



\l lit 36 1*8 

Time in hours after mixing 

two concentrations in the first determination, within the 
limits of which the end point was observed to lie. The 
process was repeated until the limit of observation was 
reached, and eventually gave two concentrations of which it 
was possible to say that one definitely caused complete 
coagulation, and one did not. The end point was taken as 
the mean of these tw^o concentrations. The observation of 
the solutions was made by daylight against a black back- 
ground. The size of the final bracket of concentration 
varied directly with the concentration of electrolyte necessary 
for coagulation, and the results were therefore more accurate 
with trivalent ions than with monovalent. The results were 
as follows (Series 5, 6 and 7, and figs. 3, 4 and 5). 



412 Mr. H. D. Murray on Influence of Size of Colloid 













CV. 




ft 


»o : : 




t^ 


^ ' 




© 






d 






ft 


o o . 
b- os • 




CO 


rn© • 




CO 






rH 






6 


OOiO 




ft 


O tp Tf 




o 


CM rH i-H 




CM 






£ 






ft 


. o »o 




l^ 






o 


• " 




cq 


















o 










o 


a 


O iO 

th © © 


Tf) 


CO 


CM CM CM 




CO 




<M 


co 




< 










■+3 


d 




d 

bf) 


& 


>Oh -f 


C 

Tf 


CM CM CM 


c, 






o 






O 


C5 




| 


ft 


b-Tf b- 




CO 


CM CM CM 


l> 


CO 




w 






W 






« 


© 




H 


ft 


OS J>. OS 


C/J 


b- 

cb 


CM CM CM 




© 






d 




















■4J 


W ^ 






S3 


^ 




O 


o r - «- 
GO rHrH 










o 






13 


,1 




£ 




-u 




<D 


e8,3 
















O 


S 2 




O 


O r~ 










OS 


o „ 






O^ 




Eh 


.1^ 






i-h 









d 






& 


iO ^ • 




CO 


OSOS • 




CO 






rH 






d 






A 


00 CO Tf 




r~ 


CO 00 O 




© 


i— 1 




CM 






d 






ft 


XHN 




© 


X) 00 OS 


0* 


Tf 










o 


















PQ 










-+■= 






d 






rB 


» 




w> 






hp 


' — ' 


CO b- Tf 


as 


CO 


OS t- OS 


o 


CO 




1 


lO 




h-H 






t> 






SQ 






H 






P3 


ft 


1Q OS CO 


rH 


b- 


OS t^- OS 


cyj 


© 

co 






d 
















m 

03 






~o 


co Eh Et< 




r- 


' v '* 




O 


CJ_ 
















03 








"IS o 










03 g 




s 






u 








o s 






o-~ 










rH 


1i q 
5 ^ 

"acq 






13 











CJ 






ft 


oo . 
o © • 






b- £~ ' 




co 






rH 






d 


©© © 




ft 


© co as 




o 


© © b- 




CM 






d 






ft 


© ©© 
rfi oo os 




»r 


iQ ir; CO 




CO 






„ 






d 






ft 


©© © 








O 


CO 

CO 


O iO © 


a 


CO 




* 






-r= 






c 






03 






,_j 


ft 


© © © 






t-^CO ^f< 


qc; 


ir~ 


HH Tf IO 


GJ 


co 




o 


tH 


1 


O 




1 


! 

i—3 


d 


! 


t— i 


ft 


. .© 


t> 


CO 


• -OS 

• -TJH 




co 




W 


ifj 










Ph 








1 


SQ 


d 


1 




ft 


© © © 
CO © IO 






Tf Tt< Tf 




CO 






CO 


I 




_d 
co 


n. B. ... 

II 

VI 




03 


W. Cm hh | 




O 


"tr — ' i 




JH 


o 




O 


c » 




-u 


O ^ : 






■J3 o ! 




■*3 


a a 




£ 


















c 






O 


o a 




, i 


Q.rH 




03 








Liminal 
NaC 





d 






p 


00 CO © 






CO CM CM 




CO 






d 






ft 


iO iO -— i 




CO 


t- Tf Tf 




00 






•^ 






d 


CO 00 rH 




Mi 


rH © © 




© 


rH 




CM 






d 






ft 


— 1 rH CM 

iC OS CO 






1—1 




CO 






CM 






d 






ft 


CO CO CM 
00 rH © 




CO 


i— 1 rH rH 




CO 






CO 




<D 






O 






CS 






«W 


o 




C3 




© © CO 


ft 


CM CO CM 


02 


© 

Tf 


CM rH rH 








c3 












O 




I 


C3 


o 




«-l-l 






^H 


ft 


CO © CO 


CV 




© IO Tf 1 


->-=> 


t~ 


CM rH rH 


d 


© 


1 


HH 


Tf 






d 


1 




ft 


rH-HTf 

© 00 © 




CO 


CO —* — 1 




co 






iO 






d 






ft 


© © Tf 

b- CM © : 




b- 


CO CM CM 




© 






© 






d 






i 














-*-: 










a 




















S 


w H 


rH ! 




o 


6^> . 




r- 






O 


CO Ph tM 




C3 














n 




















© 




o 












O 


X 




O 






P 


X 




P=l 


w 



Particles upon the Adsorption of Electrolytes. 413 

















































0^* 
































fj£ 






^a 






















^a 








































































































Cod^u 


Unt -Al 


M).- 




i 














































































IS 


Q 


J 


3o 


150 


3oo 


350 








\ 


v 


























£ 






K 


























F 


































3 


































c 






































2 






























"- 








































2 


































5 






















Fiol 










- 1 KM 




















-r 


■MAK? - 


IVaCl 












s 





. 


30 




>o 


I 


M 




50 


1 


30 









414 Size of Colloid Particles and Adsorption of Electrolytes. 

Discussion of Residis. 

It will be seen that, under the conditions imposed and 
within the limits of the experiments, a comparison of the 
data obtained, upon the basis of the total interfacial surface 
in unit volume, leads to uniformity in the curves. Such 
uniformity is not to be observed when the comparison is 
based upon the mass of the disperse phase, or the number 
of particles, in unit volume. It appears that adsorption is 
very largely conditioned by the amount of interfacial surface 
exposed. It is to be noticed, however, that the minimal 
concentration of electrolyte is higher throughout for the 
fraction containing small particles than for that containing 
large particles. This may be brought about in two ways. 
The smaller particles may bear a higher charge per unit 
area of their surface, or the critical value to which their 
charge must be reduced before coagulation begins may be 
lower than in the case of the larger particles. The latter 
explanation is more probably correct, since it is known that 
the surface tension of large particles is greater than that of 
small ones. It seems probable, if the existence of a critical 
potential difference for coagulation between a particle and 
the dispersion medium be admitted, that this should be 
lower in the case of small particles which have less tendency 
to adhere, and should thus permit of a greater freedom of 
approach between the particles. If the former explanation 
were correct, we should expect a separation of the particles 
according to size in an electric field ; but this is contrary to 
experience, the particles move at the same rate independently 
of their size. According to the Helmholtz theory of the 
electrical double la}^er, this effect is due to equal density of 
*the charge upon unit area of the surface. It appears 
probable, therefore, that the smaller particles have a lower 
critical potential difference for coagulation. The behaviour 
of the solution containing mixed particles of different size is 
in some respects curious. With both A1 2 (S0 4 ).3 and NaCl 
the curve representing the coagulation of this solution 
is more flattened relatively than the other two curves. A 
lack of uniformity in ths size of the particles appears to 
render the solution less sensitive to change in concentration, 
in the case of coagulation by univalent and trivalent ions. 



Notices respecting New Books. 415 

Summary* 

(a) A separation of the particles present in a suspension 

of gum mastic has been effected by OdeVs method 
of fractional coagulation. 

(b) The density of the particles, and the mass of mastic 

and the number of particles in unit volume have 
been measured, and from them the interfacial 
surface in unit volume calculated. 

(<•) The variation of the minimal concentrations of 
A1 2 (S0 4 ) 3 , BaCl 2 , and NaCl to coagulate solutions 
containing particles of different mean size with 
change in concentration of the solutions has been 
investigated. 

(d) It has been shown that uniformity in comparison of 
the results can be obtained upon the basis of the 
interfacial surface in unit volume. It has also been 
shown that, upon this basis of comparison, small 
particles require a higher minimal concentration of 
electrolyte than large particles. 

In conclusion I should like to thank Dr. A. S. Russell for 
his valuable advice and assistance, and Mr. H. M. Carleton 
for kindly putting at my disposal the microscopical apparatus 
required. 

Christ Church Laboratory, Oxford, 
May 1 5th, 1922. 



XLI. Notices respecting New Books. 

Basic Slags and Rock Phosphates. By Gr. Scott Robertson. 
Pp. xiv + 112, 8 plates. 1922. Cambridge Agric. Monographs. 
Cambridge University Press. 14s. net. 

r pHE value of scientific investigation of the results accruing from 
-*- the use of phosphatic dressings on crop-production is obvious to 
all, but it gains in emphasis when, as Sir E. J. Russell points out 
in a preface to the above book, agriculturists have to realize that 
the composition of basic slag has undergone much change in 
consequence of the enforced modifications in the processes of steel 
manufacture. We would go farther than Sir E. J. Russell and 



416 Notices respecting New Books. 

say that even if the war had not given an impetus to the change 
over from the basic Bessemer and acid open-hearth processes, 
economic considerations would none the less have demanded the 
development of the basic open-hearth production of steel from 
low-grade iron-ores. " This result" (to quote from the preface) 
" is, of course, distinctly awkward for the agriculturist who sees a 
valuable fertilizer disappearing, and being replaced by one which 
is more costly and at first sight seems to be nothing like as 
good." 

After a review of the various scattered experiments on the use 
of rock phosphates and basic slags hitherto undertaken, Dr. Scott 
Robertson describes in detail the Essex experiments carried out 
in the winters of 1915, 1916, 1918, and 1919 under the auspices 
of the East Anglian Institute of Agriculture. The soils treated 
were those of the Chalk, London Clay, and Boulder Clay, and 
varied considerably in mechanical and chemical composition. The 
yields of hay and clover were correlated with the rainfall, and it 
was found that the drier the season, the greater was the increase 
in production due to the use of phosphates. The botanical 
results are also given, the crowding-out of the weeds and the 
covering of bare areas with grass being noteworthy. Dr. Robert- 
son's main conclusion is that for root crops and late harvests with 
high rainfall, rock phosphates will prove a suitable substitute for 
the high-grade Bessemer basic slags. The careful records and 
correlations were made personally by Dr. Robertson at consider- 
able inconvenience and discomfort, and under most difficult 
circumstances. They are therefore the more valuable, and do him 
the greater credit. 

The latter part of the book is concerned with investigations of 
the large yields resulting from the use of basic phosphates. From 
botanical analyses it is evident that the open-hearth fluor-spar 
slags of low solubility are less effective than the non-fluor-bearing 
and therefore more highly soluble slags. The effects of the 
temperature and texture of the soil on the accumulation of 
nitrates, on the soil bacteria, and on the acidity and lime-require- 
ment are clearly expounded, and the deductions emphasized by 
means of abundant statistics. 

Altogether, the work constitutes a most valuable contribution 
to agricultural knowledge. It is a pity that the publishers 
cannot retail this book of 112 pages and 8 plates for less than 14s. 

P. G. H. B. 



Phil. Mag. Ser. 6, Vol. 44, PI. II, 

Fig. 7(a). 




Fig. 7 (b). 





Phil. Mag. Ser. 6. Vol. 44, PI. II. 

Fig. 7 («). 





Fig. 5(«). 



a -L n 

n T 




Fig. 7 (6). 




THE 
LONDON, EDINBURGH, and DUBLIN 

PHILOSOPHICAL MAGAZINE 

AND 

JOURNAL OF SCIENCE. 

[SIXTH SERIES] 



SEPTEMBER 1922. 



XLIL The Disintegration of Elements by a. Particles. By 
Sir E. Rutherford. F.R.S., Cavendish Professor oj 
Experimental Physics, and J. Chadwick, Ph.D., Clerk 
Maxwell Scholar, University of Cambridge *. 

IN a former paper f we have shown that long-range 
particles, which can be detected by their scintillations 
on a zinc-sulphide screen, are liberated from the elements 
boron, nitrogen, fluorine, sodium, aluminium, and phos- 
phorus under the bombardment of a rays. The range of 
these particles in air was greater than that of free hydrogen 
nuclei set in motion by u particles. Using radium G as a 
source of a rays, the range of the particles varied from 
40 cm. for nitrogen to 90 cm. for aluminium, while the 
range of free hydrogen nuclei under similar conditions 
was about 29 cm. 

Previous experiments J by one of us had indicated that 
the long-range particles from nitrogen were deflected in 
a magnetic field to the extent to be expected if they were 
swift hydrogen nuclei ejected from the nitrogen nucleus by 
the impinging a particle. The nature of the particles from 
the other five elements was not tested, but it seemed very 
probable that the particles were in all cases H nuclei which 
were released at different speeds depending on the nature of: 

* Communicated by the Authors. 

t Rutherford and Chad wick, Phil. Mag. vol. xlii. p. 809 (1921). 
X Rutherford, Bakerian Lecture, Proc. Roy. Soc. A, vol. xcvii. p. 374 
(1920). 

P/u7.J/a^.Ser.6.Vol.44.No.261.&?p*.1922. 2E 



418 Sir E. Rutherford and Dr. J. Chadwick on the 

the element and on the velocity of the incident a particle. 
Under the conditions of the experiment, these H nuclei 
could only arise from a disruption of the atomic nucleus by 
the action of the a particles. 

Attention was also drawn to the remarkable fact that 
in the case of the one element examined, viz. aluminium, 
the particles were liberated in all directions relative to the 
incident a particles. 

In the present paper we shall give an account of ex- 
periments to throw further light on these points and to 
test whether any evidence of artificial disintegration can 
be observed in the case of other light elements. 

Magnetic Deflexion of the Particles, 

In the course of this work, the microscope used for the 
counting of scintillations has been further improved. For 
the present experiments it was essential, in order to obtain 
a sufficient number of scintillations per minute, that the area 
of zinc-sulphide screen under observation should be greatly 
increased without diminution of the light-gathering power 
of the microscope system. Following the suggestion of 
Dr. Hartridge, a modified form of Kellner eyepiece was 
constructed. A planoconvex lens of about 7 cm. focal 
length was placed so as to render the rays of light from the 
objective approximately parallel, and the image so formed 
was viewed through an eyepiece consisting of a similar lens 
and an eye-lens of 4 cm. focal length. Used in conjunction 
with the old objective, Watson's Holoscopic of 16 mm. 
focal length and *45 numerical aperture, this system gave a 
field of view of a little more than 6 mm. diameter. A 
rectangular diaphragm was placed in the eyepiece ? limiting 
the field of view to an area 6 mm. x 4'9 mm. Our previous 
system had a field of view of 83 sq. mm. area, so that the 
new microscope, under similar conditions, gave about three 
times the number of scintillations of the old. 

The precautions adopted in counting were similar to those 
described in our previous paper. 

The method of measuring the magnetic deflexion of the 
long particles was very similar to that described by one * 
of us in the Bakerian Lecture of 1920. The experimental 
arrangement is shown in fig. 1. 

The source of a. rays was placed at R and was inclined at 
an angle of 20° to the horizontal. The lower edge was 
level with the face of a brass plate S which acted as a slit. 

* Rutherford, he. cit. 



Disintegration of Elements by u Particles. 



419 



The distance from the centre of the source to the farther 
edge of the slit was 2*95 cm. The carrier of the source and 
slit was placed in a rectangular brass box between the poles 
of an electromagnet, the field being perpendicular to the 
length of the slit. A current of dry oxygen was circulated 
through the box during the experiment. 



Fig. 1. 




If 



4- 



vS 1 

3^ 



An extension piece L, projecting 1*7 cm. beyond the edge 
of the slit, was fixed to the carrier in order to increase the 
amount of deflexion of the particles issuing from the slit. 
In the end of the box was a hole 1 cm. wide and 2 cm. long 
covered with a sheet of mica of 3*62 cm. stopping-power. 
The ZnS screen was fixed on the face of the box, leaving a 
slot of 1 mm. depth in which absorbing screens could be 
inserted. 

The source H was a brass disk of 1*2 cm. diameter coated 
with the active deposit of radium. Its initial 7-ray activity 
was usually equivalent to about 40 mgm. Ra. 

The materia], the particles from which were to be investi- 
gated, was laid directly on the source if in the form of foil, 
or if in the form of powder dusted over its face. 

The experiment consists in obtaining an estimate of the 
deflexion of the particles falling on the screen by observing 
the effect of a magnetic field on the number of scintillations 
near the line E, the edge of the undeflected beam of particles. 
The position of the microscope was fixed in the following- 
way : — After placing the source of a rays in position, 
hydrogen was passed through the box. The u rays could 
then strike the ZnS screen, and the edge E of the beam was 
clearlv defined. The microscope was adjusted so that the 

2 E 2 



420 Sir E. Rutherford and Dr. J. Chadwick on the 

edge o£ the beam of scintillations appeared a little above a 
horizontal cross-wire in the eyepiece of the microscope, 
marking the centre of the field of view. 

When the magnetic field was applied in such a way as to 
bend the a particles upwards (called the positive direction 
of the field), the edge of the beam is deflected downwards in 
the field of the microscope and the scintillations appear only 
in the lower half. When the field was applied in the opposite 
direction (negative field), the edge of the beam moved 
upwards in the field of view. The strength of the magnetic 
fields used in the experiments was always such that the 
whole field of view was covered with scintillations when 
the negative magnetic field was applied. In the experiments 
on the magnetic deflexion of the long-range particles, the 
number of particles is far too small to give a band of scintil- 
lations with a definite edge. It is clear, however, that if 
the particles are positively charged, the number of scintil- 
lations observed with the negative magnetic field will be 
greater than the number observed with the positive field, 
and that the ratio of these numbers will give a measure 
of the amount of deflexion of the particles. By determining 
this ratio for the long-range particles and comparing it with 
that for projected H particles of known velocity, we can 
obtain an approximate value for the magnetic deflexion 
of the long-range particles. The general method of the 
reduction of the observations is perhaps best shown by an 
account of the experiments on the particles from aluminium. 

Experiments on Particles from Aluminium. 

After fixing the position of the microscope in the way 
described above, an aluminium foil of 3*37 cm. stopping- 
power was placed over the source. Dry oxygen was passed 
through the box, and a mica sheet of 10 cm. stopping-power 
was inserted in front of the ZnS screen. The total absorption 
between the source and screen was then equivalent to 30 cm. 
of air. The scintillations observed were consequently due to 
long-range particles from the bombarded aluminium ; the 
ranges of the particles under observation varied from 30 cm, 
to 90 cm._, the average range being about 45 cm. 

Counts of the numbers of scintillations observed with 
positive and negative fields due to an exciting current of 
6 amps, were then made. The mean ratio of the numbers 
with a — field to those with a + field obtained from several 
experiments was 3*7. The observations were repeated with 



Disintegration of Elements by a Particles. 421 

a field due to an exciting current of 4 amps. ; the corre- 
sponding ratio was 2*1. 

When the source had decayed to a small fraction of its 
initial value, the aluminium foil over the source was removed 
and a thin sheet of paraffin wax put in its place. The mica 
sheet in front of the ZnS screen was replaced by a sheet of 
3'4 cm. stopping-power, making the total absorption equi- 
valent to 16 cm. of air. The scintillations observed on the 
screen were now due to H particles ejected from the paraffin 
wax of ranges between 16 cm. and 29 cm., the average range 
being about 22 cm. The ratio of the numbers of scintillations 
for — and + fields was determined for an exciting current of 
4 amps, and found to be 3*2. 

It appears from these results that the long-range particles 
from aluminium of average range 45 cm. were less deflected 
by the same magnetic field than H particles of average range 
22 cm. ; and that in the magnetic field due to a current of 
6 aiups.^ which was 1*34 times the intensity of the field due 
to 4 amps., they were more deflected than were the H particles 
in the latter field. To a first approximation we may say that 
the value of mvje for particles from aluminium of range 45 cm. 
is 1'23 times greater than that for H particles of range 22 cm. 
This result is clearly consistent with the view that the particles 
from aluminium are H nuclei moving with high velocity; 
for, assuming that the range of the H particle is pro- 
portional to the cube of its velocity, the velocity of a particle 
of range 45 cm. is 1*27 times that of a particle of 22 cm. 
range. 

These experiments show, therefore, that the particles from 
aluminium carry a positive charge and are deflected in a 
magnetic field to the degree to be anticipated if they are 
hydrogen nuclei moving with a velocity estimated from their 
range. While there can be little doubt that the particles 
are hydrogen nuclei, it is very difficult to prove this point 
definitely without an actual determination of the velocity and 
value of e/m of the particles. Our knowledge of the relation 
between the range and velocity of complex charged particles 
is too indefinite for purposes of calculation. On the other 
hand, if we assume, as seems a priori probable, that the 
ejected particle is the free nucleus of an atom, it is possible 
to show with some confidence that only a particle of mass 1 
and charge 1 can fit the experimental results. 

Additional evidence as to the value of mvje of the particles 
from aluminium was obtained by comparing their magnetic 
deflexion with that of the a particles of S'Q cm. range emitted 
by thorium C. In this experiment the source R was a very 



422 Sir E. Rutherford and Dr. J. Chadwick on the 

weak source of the thorium active deposit obtained by ex- 
posing a disk to thorium emanation. Hydrogen was passed 
through the box, and sufficient absorbing screens were inserted 
in front of the ZnS screen to cut out the 5 cm. a particles of 
thorium G. The numbers of a particles falling on the screen 
for — and + fields due to an exciting current of 6 amps, were 
counted, and the ratio of these numbers was found to be 2'4. 
Comparing this ratio with those found for the long-range 
particles, we see that the value of mv/e for the latter is about 
0'8 of that for a particles of 8'6 cm. range, i. e. about 
3*4 x 10 5 e.m. units. The calculated value, assuming that 
the particles are H nuclei and that their velocity is pro- 
portional to the cube root of the range, is about 3*7 X 10 5 ,e.m. 
units. Considering the difficulty of the experiments, the 
agreement is satisfactory. 

Experiments on Phosphorus and Fluorine. 

Measurements similar to the above have also been made on 
phosphorus and fluorine. 

In the case of phosphorus, a thin layer of red phosphorus 
was dusted over the face of the source. The total absorption 
in the path of the particles was about 35 cm. ; the range of 
the particles under observation varied therefore from 35 cm. 
to the maximum range of 65 cm., the average being about 
45 cm. The ratio of the numbers of scintillations for — and 
+ fields due to current of 4 amps, was 2*0. 

In the case of fluorine finely powdered calcium fluoride was 
dusted over the source. Previous experiments have shown 
that no long-range particles are emitted from calcium. The 
total absorption in the path of the particles was about 30 cm. 
The maximum range of the particles from fluorine is approxi- 
mately 65 cm., and the average range of the particles falling 
on the screen was around 40 cm. The ratio of the numbers 
of particles observed for — and ■+■ fields due to an exciting 
current of 4 amperes was 2*5. 

It is clear from these results that, within the error of 
experiment, the particles liberated from phosphorus and from 
fluorine are bent in a magnetic field to approximately the 
same extent as the particles from aluminium. We may 
conclude, therefore, that these particles also are H nuclei 
moving with high speed. 

We have not examined the particles from boron and 
sodium in this way, but there seems no reason to doubt that 
they also consist of H nuclei. 



Disintegration of Elements by a Particles. 



423 



The Ranges of the H Particles. 

In the experiments described in our previous paper only 
two elements, nitrogen and aluminium, were investigated 
in any detail. The other elements were examined in a 
qualitative manner, but it was shown that the ranges of 
the liberated H particles were in every case greater than 
40 cm. of air. The ranges of the particles from these 
elements — viz. boron, fluorine, sodium, and phosphorus — 
have now been determined more accurately. 

Attention has been drawn to the remarkable fact that the 
H particles liberated from aluminium appeared not only in 
the direction of the incident ot particles but also in the reverse 
direction. The number of particles emitted in the backward 
direction was of the same order of magnitude as for the 
forward, but the maximum range in the backward direction 
was smaller, being 67 cm. as against the 90 cm. range of the 
forward particles, for a particles of 7 cm. range. Some 
experiments with nitrogen showed that the number of 
H particles emitted in the backward direction was very 
small at absorptions of more than 18 cm. of air. 

We have repeated these experiments and extended them 
to include the other elements boron, fluorine, sodium, and 
phosphorus, with the result that we find that in every case 
the H particles emitted on disintegration of the nucleus 
escape in all directions, the maximum range in the backward 
beino- less than in the forward direction. 



Fig. 2. 



A 



A 



BLJ<t0 



!l!ljllll|lt!l|l||i|lll 

The experimental arrangement for the measurement of 
the ranges of the forward particles was the same as that 
described in our previous paper. The apparatus used in 
the investigation of the particles in the reverse direction 
differed from this in the arrangement of the source, and is 
shown in the diagram (fig. 2). 

The source of a particles was carried on a rod passing with 
a sliding fit through a stopper which fitted tightly into the 



424 Sir E. Rutherford and Dr. J. Chad wick on the 

brass tube T of 3 cm. diameter. The end of this tube was 
provided with a hole 7*5 mm. in diameter, closed by a silver 
foil of 3*75 cm. air equivalent. The zinc sulphide screen S 
was fixed on the face of the vessel leaving a slot in which 
absorbing screens could be inserted. The apparatus was 
placed between the poles of an electromagnet to reduce the 
luminosity produced in the screen by the /3 rays. 

The source R was a silver foil of 4'15 cm. stopping-power 
coated on one side only with the active deposit of radium. 
Its initial 7-ray activity was in most experiments equiva- 
lent to about 30 mg. Ra. The inactive side of the silver 
foil faced towards the ZnS screen. The distance of the source 
from the screen was generally about 3*5 cm., but could be 
varied, and its position read off on a scale. The elements 
to be examined could in most cases only be obtained in 
the combined state. The powdered compound was heated 
in vacuo, and a film prepared by dusting on to a foil smeared 
with alcohol. The screen thus prepared was placed immedi- 
ately behind the source. As in our previous experiments, a 
stream of dry oxygen was circulated through the apparatus. 

In all cases, except that of nitrogen, the maximum range 
of the particles emitted in the backward direction was 
greater than the range of free hydrogen particles, so that no 
complication arises from the presence of hydrogen in the 
silver foils or other materials in the path of the a particles. 

In the case of nitrogen, however, as our previous experi- 
ments had shown, the range of the backward particles is 
much less than that of free hydrogen particles, and it was 
consequently necessary to allow for the " natural" effect, i. e. 
for the H particles arising from hydrogen contamination of 
the source and screens in the path of the a rays. It was 
found inconvenient to use gaseous nitrogen for these experi- 
ments, and a suitable screen was prepared by sifting a thin 
layer of powdered paracyanogen, CyST x , on to a gold foil. 
The scintillations observed on the ZnS screen when the film 
of paracyanogen was placed against the source were due to 
the " natural " particles from the source and screens, together 
with those which came from the nitrogen in the paracyanogen. 
On taking away the film of C Z N X the natural particles alone 
were counted. In some experiments a film of paraffin wax 
was placed against the source. The natural effect remained 
the same, showing that even if the film of paracyanogen 
contained a large amount of hydrogen the number of free H 
particles scattered to the ZnS screen by the walls of the 
vessel was negligible. 

Figure 3 shows the type of results obtained in these 



Disintegration of Elements by « Particles. 



425 



experiments. The ordinates represent the number o£ scintil- 
lations observed per minute per milligram of activity of radium 
C, measured by 7 rays ; the abscissae, the stopping power 
for a. rays of the absorbing screens, expressed in terms of 
centimetres of air. The dotted curve A gives the natural 
effect observed when the screen of paracyanogen was absent 
or replaced by a film of paraffin wax ; the full curve B the 
effect when present. The difference of these curves there- 
fore represents the effect due to the nitrogen in the para- 
cyanogen. It will be seen that the maximum range of the 



Fig-. 3. 



20 



1-5 



0-5 



\ 




Forward and Ba 
from Afitt 


cAward Particles 
■oyen. 








\ 














~-JV 


\B 








\cC 








■ -»» 


% 









:0 25 30 

A bsorption m cms of sir 



40 



backward particles from nitrogen is about 18 cm. Curve C 
is the absorption curve for the particles emitted by nitrogen 
in the forward direction. 

In the following table are given the maximum ranges of 
the particles liberated from the elements which show the 
disintegration effect, for both forward and backward 
directions. 

Element. 



Forward rang* 



Boron 58 

Nitrogen 40 

Fluorine 65 

Sodium 58 

Aluminium 90 

Phosphorus ••• 65 



Backward range. 
cm. 

38 
18 
48 
36 
67 
49 



426 Sir E. Rutherford and Dr. J. Chadwick on the 

It should be pointed out that the ranges of the forward 
particles from boron, fluorine, sodium, and phosphorus may 
be subject to considerable error, owing to the use of a film of 
powder as the bombarded material. The particles of 
maximum range are produced on the surface of the grains of 
powder, and therefore to find the true range the size and air 
equivalent of the grains of powder must be known. For the 
ranges given above it has been assumed that the grains were 
uniform in size and an average value of the air equivalent of 
the film of powder has been calculated from its weight per 
sq. cm. The ranges so determined are obviously somewhat 
less than the true ranges. The ranges of the backward 
particles are, of course, not subject to this source of error. 

It was observed that the number of particles liberated 
from the different elements appeared all to be of the same 
order of magnitude when allowance is made for the differ- 
ence in range. In our original experiments we found that 
the number ojj particles from boron was somewhat smaller 
than the numbers from the other elements, but this was 
due to the use of an irregular film. Using a film of more 
finely powdered boron it was found that the number of 
particles from boron was about the same as from the other 
elements. 

Examination of other Elements. 

In our former experiments we examined all the light 
elements, with the exception of the rare gases, as far as 
calcium. Of these only the six elements of the above table 
were found to emit H particles in detectable amount under 
the bombardment of & rays. As was pointed out in that 
paper, the atomic masses of these elements can be represented 
by 4n + a where n is a whole number, a result which 
receives a simple explanation on the assumption that the 
nuclei of these elements are composed of helium nuclei of 
mass 4 and hydrogen nuclei. On the other hand, some 
of the light elements which gave no detectable number of H 
particles also had atomic masses given by 4w + a. It was 
thus a point of great importance to repeat the examination of 
these elements with the improved microscope, and to search, 
if possible, for the emission of particles of shorter range than 
free H nuclei. In some cases it was only possible, on account 
of hydrogen contamination of the materials, to observe at 
absorptions greater than 30 cm. of air, while in others the 
observations were carried well within this range. 

Lithium was examined as oxide and as metal, a thin sheet 
of the latter being obtained by pressing molten lithium 



Disintegration of Elements by a Particles. 



427 



between two steel plates in an atmosphere of carbon dioxide. 
No evidence was found of any particles of range greater than 
30 cm. Owing to hydrogen contamination of the Li and 
Li 2 the observations at smaller ranges were not decisive. 
Observations in the backward direction revealed no detectable 
number of particles of range greater than 14 cm. 

Beryllium was examined as the powdered oxide, and there 
was again no evidence of the emission of particles of longer 
range than 30 cm. in the forward direction or 15 cm. in the 
backward. 

Magnesium was examined with a sheet of the metal and 
also with a screen of powdered magnesium. There was no 
evidence of long-range particles. 

For silicon a screen of powdered silicon and a thin sheet 
of quartz were used. With the sheet of quartz it was 
possible to make observations in the forward direction at 
absorptions as low as 17 cm. The scintillations observed 
were due entirely to the natural H particles. 



To CO, 




Chlorine had been previously examined in the form of 
various chlorides. These observations were repeated, and the 
results confirmed the conclusion that particles of greater 
range than 30 cm. were not liberated in any detectable 
amount. In order to pursue the observations within the 
range of free H particles a special series of experiments was 



428 Sir E. Rutherford and Dr. J. Chadwick on the 

carried out. A glass apparatus, similar in design to the 
standard apparatus, was used. The details will be clear from 
the diagram (fig. 4). 

In order to avoid the bombardment of the glass walls and 
consequent liberation of H particles the inside of the tube 
was lined with platinum foil. The surfaces of the brass 
plate B and of the rod carrying the source were protected 
from the action of the chlorine by a coating of hard pitch. 
The stopcocks and grouod-joint were lubricated with a 
brominated grease. The source of a rays was a platinum foil 
coated with radium active deposit. Pure dry chlorine was 
prepared by heating gold chloride, AuCl 3 , contained in the tube 
A, and was passed over P 2 5 before entering the vessel T. 
As an additional precaution a little P 2 0$ was placed in the 
vessel itself. 

When the source was placed in position the air was 
removed by pumping and washing with dry carbon dioxide. 
Carbon dioxide was then let in to atmospheric pressure and 
the natural H particles were counted at absorptions varying 
from 16 cm. to 30 cm. The carbon dioxide was then 
replaced by chlorine, and the scintillations at similar absorp- 
tions were observed. The chlorine was then allowed to be 
reabsorbed by the gold chloride and carbon dioxide let in 
again. In this way counts on the chlorine were included 
between counts of the natural particles, and any traces of 
adventitious hydrogen could be allowed for. The results 
showed no evidence of the liberation of H particles from 
chlorine in the range examined, i. e. at absorptions more than 
16 cm. of air. 



Discussion of Results. 

For convenience of discussion the atomic numbers and the 
masses of the isotopes of the elements from hydrogen to 
potassium are given in the following table. Of these 
elements aluminium is the only one which has not yet been 
examined for isotopes, but it appears likely that it is a pure 
element of atomic mass 27. With the exception of helium, 
neon, and argon, all the elements in the table have been 
tested to see whether H nuclei are ejected by the action, of a 
particles. The six active elements, as they may be termed 
for convenience, are underlined. 



Disintegration of Elements by a Particles. 
Table T. 



429 



Element. £ t01 » ic 
a amber. 

h r~ 

He 2 

Li 3 

Be 4 

B 5 

c" 6 

N 7 

O 8 

Fl 9 

Ne 10 



Atomic 
Masses. 



1-008 

400 

6,7 

9 
10,11 
12 
14 
16 
19 
20,22 



Element. ,£ toD ? io 
.Number. 

2ii IT - 

Mg 12 

Al 13 

Si 14 

P 15 

S 16 

CI 17 

A 18 

K 19 



Atomic 
Masses. 



23 

24, 25. 26 

27 
28.29 

31 

32 
35,37 
36,40 
39,41 



An examination of the table shows that the active elements 
may be classified in different ways : — 

(1) Active elements are odd-numbered elements in a 

regular sequence of numbers, viz., 5, 7, 9, 11, 13, 15. 

(2) The atomic masses of the active elements are given by 

472 + a where n is a whole number ; a = 3 for all the 
elements except nitrogen, for which it is 2. 

(3) With the exception of boron, which has two isotopes 

(10, 11), the active elements are all pure elements. 

We have seen that no evidence has been obtained that the 
preceding element lithium (3), and the succeeding elements, 
chlorine (17) and potassium (19), show any trace of activity 
under a-ray bombardment, although they are odd-numbered 
elements and the masses of their isotopes are given by 4n-f- a. 
Magnesium and silicon, which are even-numbered, but which 
contain isotopes of mass 4?i -t- 1 or 4n + 2, show no sign of 
activity. 

There thus appears to be no obvious general relation which 
differentiates active from inactive elements. The activity 
starts sharply with boron and ends abruptly with phosphorus. 
It is a very unexpected observation that neither lithium nor 
chlorine shows any certain evidence of activity in the emission 
of either long-range or short-range particles. It is of 
interest to consider whether any deduction can be made as to 
the structure of these nuclei in the light of these experimental 
facts. 

In our previous paper it was pointed out that the H nuclei 



430 Sir E. Rutherford and Dr. J. Chadwick on the 

liberated from the active elements probably existed as 
satellites circulating in orbits round the main nucleus. In 
the case of an effective collision of an a particle with such 
a nucleus, part of the momentum of the a particle is com- 
municated to the central nucleus, but the satellite is 
sufficiently distant from the latter to acquire enough 
momentum and energy to escape from the system. It was 
shown that such a point of view offers a general explana- 
tion of the variation of the velocity of the expelled H 
nuclei with the speed of the a particle and also of the escape 
of the H nuclei in all directions. The chance of ejecting 
an H satellite at high speed from a nucleus is much smaller 
(for nitrogen, for example, about 1/20) than the chance of 
setting a free H nucleus in correspondingly rapid motion. 
It appears therefore that the release of the satellite only 
takes place under certain restricted conditions of the 
collision of the v particle with the nucleus. If the H 
satellites were present in lithium and chlorine and were very 
lightly bound to the nucleus, it is to be anticipated that 
the number released by the a rays would be of the same 
order of magnitude as if the H nuclei were free. As this 
is found not to be the case, we may conclude that neither 
lithium nor chlorine has any lightly bound satellites in its 
nuclear structure. The complete absence of long-range 
particles from these elements shows that the H satellites, if 
they are present at all, are strongly bound to the main 
nucleus. If, for example, the satellite revolves very close to 
the nucleus, the a particle may only be able to give such a 
small part of its momentum to the satellite that it is unable 
to release it from the system. It does not, however, seem 
likely that the forces binding a satellite would vary greatly 
in passing from phosphorus to chlorine. It seems more 
probable that the general structure of the chlorine nucleus 
differs in some marked way from that of the group of active 
elements. The H nuclei may perhaps be definitely incor- 
porated into the main nuclear system, so that the a particle 
has no chance of concentrating its energy upon a single unit 
of the nuclear structure. In a similar way it seems probable 
that lithium must differ widely in structure from the suc- 
ceeding element boron. The facts brought to light in these 
experiments indicate that the nuclei even of light elements 
are very complex systems and illustrate how difficult it will 
be to find any simple and general rule to account for the 
variation in structure of successive elements. 

It has been pointed out that, with the exception of the first 



Disintegration of Elements by a. Particles. 431 

element boron, all the active elements are " pure " elements, 
i. e., have no isotopes. This may be of some significance in 
differentiating between the structure of active and inactive 
elements. The absence of isotopes indicates that, as regards 
mass, there is only a narrow range of stability of the nucleus 
for a given nuclear charge ; the addition or subtraction of 
an equal number of H nuclei and electrons leads presumably 
to an instability of the nuclear system. In the case of 
lithium and of chlorine, which form isotopes, the forces 
binding the nuclei together may consequently be very 
different from those in the case of the pure active elements. 
If there is any significance in this point of view, it would 
indicate that H satellites are only present in pure odd- 
numbered elements ; but, as we have seen, boron is an 
exception to this rule. 

In comparing the phenomena shown by the six active 
elements, it seems at once clear that nitrogen occupies an 
exceptional position in the group. Not only is the range of 
the expelled H nuclei the smallest of all the group, but the 
ratio of the ranges in the two directions is markedly different 
from those shown by the other elements. It is natural to 
connect this anomalous behaviour with the fact that the mass 
of the nitrogen nucleus is given by 4w + 2, while the rest of 
the group are of the class 4n + 3. The slower speed of 
ejection of the particles from nitrogen at first sig;ht suggests 
that the H satellite is more lightly bound than in the case of 
the other elements. This suggestion is, however, not borne 
out by calculation of the distribution of momentum among 
the three bodies involved in the collision, viz., the a particle, 
the H satellite, and the residual nucleus. Jn our previous 
paper, we showed that the distribution of momentum could 
be calculated on certain assumptions from the observations 
of the ranges of the expelled nuclei in the forward and 
reverse directions of the a particle. It was supposed that 
the law of conservation of momentum holds, and that the sum 
of the energies of the H particle and the residual nucleus 
was the same whether the H particle was liberated in the 
forward or backward direction. It follows from these 
assumptions that the relative velocity of the H nucleus and 
the residual nucleus is the same in the two cases. The results 
of this calculation for the group of active elements are 
collected in the following table (Table II.). 



-54 Y 


323 Y 


2-52 Y 


42% 


1-44 V 


454 Y 


•78 V 


-13 % 


--10Y 


3-89 Y 


2-00 Y 


35 % 


1-42 Y 


5-16 Y 


•56 Y 


6% 


•78 Y 


523 Y 


•88 Y 


42% 



432 The Disintegration of Elements by ol Particles. 

Table II. 
Distribution of Momentum. 

H particle. Residual Nucleus. a Gain in 

me " Forward. .Backward. Forward. Backward, particle. Energy- 

Boron 202 Y - 1'75 Y 

Nitrogen 1*78 V -1'32 Y 

Fluorine 2-10 Y - T89 Y 

Sodium 202 Y -172 Y 

Aluminium... 234 V -211 Y 

Phosphorus... 210 Y -1-89 V VUT 5-13 Y 76 Y 15% 

The momenta are expressed in terms of the initial velocity 
V of the a particle. The initial momentum of the a particle, 
and consequently the sum of the momenta of the three bodies 
after collision, is therefore 4V. Momenta in the direction of 
the incident a particle are taken as positive, momenta in ths 
opposite direction as negative. The percentage energy, 
gained from the nucleus as a result of the disintegration, is 
given in the last column, in terms of the initial energy of the 
ol particle. 

It will be seen that in the case of nitrogen a considerable 
part of the momentum of the a particle is communicated to 
the main nucleus, a much greater part than in the cases of 
the adjacent elements boron and fluorine. This indicates 
that the H satellite of nitrogen is in relatively close 
proximity to the main nucleus. It will also be noted that 
while for the other elements there is a gain of energy from 
the disruption varying from 6 per cent, for sodium to 42 per 
cent, for boron and aluminium, for nitrogen there is a loss of 
energy of 13 per cent. 

It is apparent from the above table that the distribution 
of momentum among the three bodies varies considerably for 
the different elements, but, in the absence of any definite 
evidence of the validity of the theory on which the calcula- 
tions are based, it seems inadvisable to discuss these differences 
in any detail at the present stage. 

Cavendish Laboratory, 
June 20, 1922. 



L 433 ] 



XLIII. The Distribution of Electrons around the Nucleus in the 
Sod/ ion and Chlomne Atoms. By W. Lawren'CE Bragg, 
M.A., T.lx.S.. Langworthy Professor of Physics, The 
University of Manchester ; R. W. James, M.A., Senior 
Lecturer in Physics, The University of Manchester ; and 
C. H. Bosanquet, M. A., Balliol College, Oxford*. 

1. TN two recent papers f in the Philosophical Magazine 
i the authors have published the results of measure- 
ments made on the intensity of: reflexion of X-rays by 
rock-salt. The mathematical formula for the intensity of 
reflexion, as calculated by Darwin J, involves as one of 
its factors the amount of radiant energy scattered in various 
directions by a single atom when X-rays of given amplitude 
fall upon it. The other factors in the formula can bo 
evaluated. By measuring the intensity of reflexion experi- 
mentally we can therefore obtain an absolute measurement 
of the amplitude of the wave, scattered by a single atom, 
in terms of the amplitude of the incident radiation. 

This measurement is of considerable interest, because it 
may throw some light on the distribution of the electrons 
around the nucleus of the atom. We regard the wave scat- 
tered by the atom, as a whole, as the resultant of a number of 
waves, each scattered independently by the electrons in the 
atom. A formula first evaluated by J. J. Thomson is used 
in order to calculate the amplitude of the wave scattered by 
a single electron. If an incident beam of plane polarized 
X-rays consists of waves of amplitude A, then the amplitude 
A' at a distance R from the electron in a plane containing 
the direction of the incident radiation, and at right angles 
to the electric displacement, is given by 

A R mc 2 {) 

Here e and m are the charge anc [ m ass of the electron in 
electromagnetic units, and c is the velocity of light. 

What we measure experimentally is the resultant ampli- 
tude of the wave-train scattered in various directions by a 
number Z of electrons in the atom. If all the electrons were 

* Communicated by the Authors. 

t Phil. Mag. vol. xli. March 1921 ; vol. xlii. July 1921. 
X C. G. Darwin, Phil. Mag. vol. xxvii. pp. 315-675 (Feb. and April 
1914). 

Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 F 



434 Prof. W. L. Bragg and Messrs. James and Bosanquet : 

concentrated in a region whose dimensions were small com- 
pared with the wave-length of the rays, then the resultant 

Z e 2 
amplitude would be equal to ^ — ^ smce the scattered 

_tx Tnc 

wavelets would be in phase with each other in all directions. 

It is found experimentally that the measured amplitude 

tends to a value which is in agreement with the formula 

at small angles of scattering, but that at greater angles it 

falls to a very much smaller value. This is to be accounted 

for by the action of interference between the waves scattered 

by the electrons in an atom, which are distributed throughout 

a region whose dimensions are large compared with the 

X-ray wave-length. 

It is an easy matter to calculate the average amplitude 
scattered in any direction by a given distribution of electrons 
around the nucleus. Here we are attempting to solve the 
reverse of this problem. The experimental results tell the 
amplitude of the wave scattered by the sodium and chlorine 
atoms through angles between 10° and 60°. We wish to 
use these results in order to get some idea of the manner 
in which the electrons are distributed. 

2. In addition to Darwin's original mathematical treat- 
ment, the question of the effect on X-ray reflexion of the 
distribution of electrons around the atom has been dealt 
with by W. H. Bragg *, A. H. Compton f , and P. Debye 
and P. Scherrerif. 

W. H. Bragg considered the interpretation of the diminu- 
tion in the intensities of reflexion by a crystal as the 
glancing angle is increased, due allowance being made for 
the arrangement of the atoms. He concluded that "an 
ample explanation of the rapid diminution of intensities 
•is to be found in the highly probable hypothesis that the 
scattering power of the atom is not localized at one central 
point in each, but is distributed through the volume of the 
atom." He did not regard the experimental data then 
available as sufficient to justify making an estimate of the 
distribution of the electrons. These data indicated that the 

intensity of reflexion fell off roughly as . 2 ~ (0 being 

the glancing angle), and he showed that a density of 
distribution of the electrons could be postulated which 

* W. H. Bragg, Phil. Trans. Roy. Soc. Series A, vol. ccxv. 
pp. 253-274, July 1915. 

t A. H. Compton, Phys. Rev. vol. ix. no. 1, Jan. 1917. 

t P. Debye and P. Scherrer, Phys, Zeit. pp. 474-483, July 1918. 



Distribution of Electrons in JSa and CI Atoms. 4IJ5 

accounted for this law, just as an illustration of the appli- 
cation of the principle involved in considering spatial 
distribution. 

A. H. Compton used the experimental results obtained by 
W. H. Bragg in order to calculate the electron distribution. 
W. H. Bragg showed that the intensity of reflexion is a 
function of the angle of reflexion alone, when allowance has 
been made for the arrangement of the atoms in the crystal, 
and he determined the relative intensity of reflexion by a 
number of planes in rock-salt and calcite. Compton cal- 
culated from these values the relative amplitudes of the 
waves scattered by the atoms in different directions, by 
means of the reflexion formula of Darwin, and proceeded 
to test various arrangements of electrons in order to find 
one which gave a scattering curve agreeing with that found 
experimentally. He supposed that the electrons were 
rotating in rings, governed by Bohr quantum relationships 
In sodium, for example, he placed four electrons on an 
inner ring, six on the next ring, and a single valency electron 
on an outer ring. In chlorine the rings contained four, six, 
and seven electrons respectively. Compton found that these 
atomic models gave a fair agreement with W. H. Bragg's 
results. 

Debye and Scherrer came to the same conclusion as to the 
significance of intensities as regards electron distribution 
which was implied in W. H. Bragg's work and stated more 
fully by Compton. They considered two interesting cases. 
The first was that of the lithium fluoride crystal. They 
compared the intensity of reflexion by planes where the 
fluorine and lithium atoms reflected waves in phase with 
each other, with that by planes where these atoms acted 
in opposition to each other. The relative amplitudes at any 

TT -i- T i 
angle for such planes may be expressed by the ratio ^ — ^> 

where F and Li are the amplitudes contributed by the fluorine 
and lithium atoms respectively. Their figures indicated 

Ui r • 

that the limiting values of ^ — p at zero angle of scattering 

is 1*5, signifying that a valency electron has passed from the 

lithium to the fluorine atom I ^ =1*5 ). 

Their intensities of reflexion were measured by the 
darkening of a photographic plate in the powder method 
of analysis which these authors initiated. In view of the 

2F2 



436 Prof. W. L. Bragg and Messrs. James and Bosanquet : 

difficulties o£ estimating intensities in this way, of the few 

points which they obtained on the curve for the ™ x • ratio, 

of the difficulties in interpreting intensities which we have 
discussed in our papers, and of the large extrapolation 
which they had to make in order to get the limiting value 

of -j= — j-. , we feel that their results cannot be regarded as 
r — .Li 

proving that the transference of the valency electron has 

taken place. The fact of the transference is supported by 

much indirect evidence, and their conclusion is probably 

correct. 

Debye and Scherrer also compared the intensities reflected 
by various planes of the diamond, and concluded that the 
electrons in the carbon o atoms were contained within a 
sphere of diameter 043 A, assuming a uniform distribution 
throughout this sphere. 

In all the above cases, the results were obtained by com- 
paring the relative intensities of reflexion by various faces. 
The results which we have obtained, and which will be used 
to calculate the distribution of electrons in sodium and 
chlorine, are, on the other hand, absolute determinations. 
The intensity of reflexion was compared in each case with 
the strength of the primary beam of X-rays, so that the 
absolute efficiency of the atom as a scattering agent could 
be deduced. 

In a paper on "The Reflection Coefficient of Monochro- 
matic X Rays from Bock Salt and Calcite " *, Compton 
made comparisons of the incident and reflected beam, for 
the first order reflexion from cleavage faces of these crystals. 
He obtained results for rock-salt which were rather less than 
those which we afterwards obtained for a ground face, but 
he noted that the effect was increased by grinding the face. 
In our notation the results were 

Compton ^ - -00044 + '00002 

;XaCl(100). 
B.J. and B. ^ = v 00055 

As Compton surmised, and as we have found experimentally, 
this figure for the efficiency of reflexion has to be modified 
considerably to allow for the extinction factor. The difference 

* A. H. Compton, Phys. Rev. vol. x. p. 95, July 1917. 



Distribution of Electrons in Na and CI Atoms. 437 

between his results and ours is accounted for by the extinction 
or increased absorption of the rays at the reflecting angle. 
Compton pointed out that the reflexion factor was of the 
order to be expected from Darwin's formula, but did not 
use the value he obtained to solve the electron-distribution 
problem. 

3. For the sake of convenience of reference, the formula 
which forms the basis of all the calculations is quoted below. 
Let the intensity I of a beam of homogeneous X-rays, at a 
given point, be defined as the total energy of radiation falling 
per second on an area of one square centimetre at right 
angles to the direction of the beam. If a crystal element of 
volume dV, supposed to be so small that absorption of the 
rays by the crystal is inappreciable, be placed so that it is 
bathed by the X-rays, and if it is turned with angular 
velocity co through the angle at which some plane in it 
reflects the X-rays about an axis parallel to that plane, the 
theoretical expression for the total quantity of energy of 
radiation E reflected states that 

■^ = NV F «.4- 4 i±^.-B-n-»iV . (2) 

I sin 2d m 2 c 4 2 v ' 

= QdV. 
In this expression 

N= Number of diffracting units per unit volume*. 

A, = Wave-length of X-rays. 

6 = Glancing angle at which reflexion takes place. 

e = Electronic charge. 

m = Electronic mass. 

c = Velocity of light. 

The factor e~ Bsm " (the Debye factor) represents the effect 
of the thermal agitation of the atoms in reducing the 
intensity of reflexion. 

The factor F depends on the number and arrangement of 
the electrons in the diffracting unit. At = it would have 
a maximum value equal to the total number of electrons in 
the unit, and it falls off owing to interference as 6 increases. 

The experimental observations have as their object the 
determination of Q in absolute units. In practice we cannot 
use a single perfect crystal so small that absorption is 

* Xo account is taken here of the " structure factor." The diffracting 
units are supposed to be spherically symmetrical as regards their 
diffraction effects. 



438 Prof. W. L. Bragg and Messrs. James and Bosanquet r • 

inappreciable. We use a large crystal consisting of num- 
bers of such homogeneous units and deduce, from its 
reflecting power, the reflecting power Q per unit volume 
of the units of which it is composed. The assumptions made 
in doing this are by no means free from objections, and will 
be discussed later in this paper. Taking this to be justifiable, 
however, our experimental results yield the value of Q for 
rock-salt over a wide range of angles, and from them the 
values of F C i and F Na follow directly. These values are 
shown in fig. 1. 

Fur. 1. 



18 




1 










\ 
\ 

\ 
% 












12 

8 


\ 

\ 
'I 












\ 
















































4 
























*P) 














'(c) 



0-3 
S/o A 



(a) F C1 corrected for Debye factor, (c) F Na corrected for Debye factor. 

(b) F cl uncorrected „ „ (d) F Na uncorrected „ ,, 

4. We must now consider more closely the significance of 
the factor F. The most simple case is that of a crystal con- 
taining atoms of one kind only. Parallel to any face of the 



Distribution of Electrons in Na and CI Atoms. 439 

crystal we can suppose the atoms all to lie in a series of 
planes, successive planes being separated by a distance d. 
We get the nth order spectrum formed at a glancing angle 
by the reflexion from such a set of planes if 

2d sin = n\. 

This spectrum represents the radiation diffracted by the 
atoms in a direction making an angle 26 with the incident 
beam, and it is formed because in this particular direction 
the radiation scattered by any pair of atoms lying in suc- 
cessive planes differs in phase by 2mr. Thus the amplitude 
of the beam scattered in this direction is the sum of the 
amplitudes scattered by all the neighbouring atoms taking 
part in the reflexion. 

Let us consider the contribution to the reflected beam of 
a group of atoms lying in a reflecting plane. To obtain the 
amplitudes of the reflected wave, we sum up the amplitudes 
contributed by the electrons in all the atoms, taking due 
account of the fact that the electrons do not in general lie 
exactly in the reflecting plane and so contribute waves 
which are not in phase with the resultant reflected wave. 
By symmetry, the phase of the resultant wave will be the 
same as that reflected by electrons lying exactly in the 
geometrical plane passing through the mean positions of all 
these atomic centres. The phase of the wave scattered in a 
direction 6 by an electron at a distance x from the plane 
differs from that of the resultant wave by an amount 

— -.2?sin 6. 
A, 

We will suppose that there is in every atom an electron 
which is at a distance a from the centre, and that all direc- 
tions of the radius joining the electron to the atomic centre 
are equally likely to occur in the crystal. In finding the 
effect of these electrons for all atoms (M in number) of the 
group, we may take it as equivalent to that of M electrons 
distributed equally over a sphere of radius a. It can easily 
be shown that, if x is the distance of an electron from the 
plane, all values of x between +a and —a are equally likely 
for both cases. Such a shell scatters a wave which is less 
than that scattered by M electrons in the plane in the ratio 

sin 6 , 
— — -, where 

* 4tt 

(/)= --a sin 0. 



440 Prof. W. L. Bragg and Messrs. James and Bosanquet : 

The average contribution of the electron in each atom to the 

F factor is therefore —7— , and not unity as it would be 

if the electron were at the centre of the atom. 

If there are n electrons at a distance a from the centre of 
the atom, their contribution to the F factor would be 

sin 6 , Q . 

n -f < 3 > 

Any arrangement of n electrons at a distance a from the 
centre of the atom, provided that all orientations of the 
arrangement were equally probable, would make the same 
contribution to the F factor. Foe example, eight electrons 
arranged in a ring about the nucleus would give the same 
value for F as eight electrons arranged at the corners of a 
cube, or eight electrons rotating in orbits lying on a sphere 
of radius a. This illustrates the limitations of our analysis, 
which cannot distinguish between these cases. We can only 
expect to get information from our experimental results as 
to the average distance of the electrons from the atomic 
centre, and this for the average atom. 

Suppose now that any atom contains a electrons at a 
distance ?\ from the nucleus, b at a distance r 2 , c at a dis- 
tance r B . . . n at a distance r n , then the value of F for the 
average atom would be given by 

F =a !i^ + 6 si ^ + o%i 3 + ... + n ^i". . (4) 
9i 92 93 9« 

Thus, given the distribution of the electrons on a series of 
shells or rings, we can calculate the value of F for any value 
of 6. The problem we have to solve here, however, is the 
converse of this. We have measured the value of F for a 
series of values of #, and wish to determine from the results 
the distribution of the electrons. We have seen above that 
there is no unique solution of this problem, but we can get 
some idea of the type of distribution which will fit the 
experimental curves. 

In order to do this, we suppose the electrons to lie on a 
series of shells, of definite radii r 1; r 2 , .... and determine 
the number of electrons a, b, c on the various shells which 
will give values of F corresponding to those observed 
experimentally. Suppose, for example, we take six shells 
uniformly spaced over a distance somewhat greater than 



Distribution of Electrons in Na and CI Atoms. 441 

the atomic radius is expected to be. For any given value 
of 6 we have 

</>l </>2 </>3 </>t $5 

+/^ (5) 

We chose from the experimental curve six values of 6 
evenly spaced over the range of values at our disposal, and 
for each of these values read from the curve the value of F. 
Since definite radii have been assumed for the shells, the 

values of — j-^~ , etc., can be calculated for each value of 6. 
9i 

Hence, for each value of 0, we have an equation involving 
numerical coefficients and the quantities a, b, c, d, e,f\ so that 
if six such equations are formed we may calculate these 
quantities. 

If Z is the total number of electrons in the atom w r e have 

Z=a+b + c + d + e+f, ...... (6) 

and this will be taken as one of our equations (corresponding 
to # — 0). In calculating the results for sodium and chlorine 
we have assumed the atom to be ionized, and have taken 
Zci = 18 and Z^ a = 10. 

It will be evident that this method of solution is somewhat 
arbitrary, and that the results we get will depend on the 
particular radii assumed for the shells. By assuming various 
radii for the shells, however, and solving the simultaneous 
equations for the number of electrons on each, we find that 
the solutions agree in the number of electrons assigned to 
various regions of the atom. 

As a test of the method of analysis, a model atom was 
taken which w T as supposed to have electrons arranged as 
follows : — 

2 on a shell 0"05 A radius. 
5 „ „ 0-35 

3 „ „ 0-70 

The F curve for this model was calculated. Then the simul- 
taneous equations for the electron distribution were solved, 
just as if this curve had been one found experimentally. 
This was done for two arbitrarily chosen sets of radii, taken 
out to well beyond the shell at 0*70 A. 






442 Prof. W. L. Bragg and Messrs. James and Bosanquet : 

The comparison between the two analyses (dotted curves) 
and the atom model we started with (continuous curve) is 
shown in fig. 2. The abscissas represent the radii of the 
shells in A, the ordinates the total number of electrons 
inside a shell of that radius. When the limits of the atomic 
structure are reached, the curve becomes horizontal at the 
value 10, corresponding to the ten electrons. The analyses 
not only indicate with considerable accuracy the way in 



n3 
fl 4 



Fig. 2. 



xr._ 



1-5 

sphere, measured 



2-0 
Angstrom units. 



2-3 



which the electron-content grows as we pass to spheres of 
larger radii, but also tell definitely the outer boundary 
of the atomic structure. Both give a number of electrons 
very nearly equal to zero in the shells outside O70 A. 

5. The F curves for sodium and chlorine can be solved 
in the same manner. We have expressed our results in 
two ways. 

First, we have supposed the electrons to be grouped on 
shells. The numbers of electrons on each shell, and the 
radii of the shells, have been so adjusted as to give the best 
possible fit to the experimental curves. In the case of 
sodium it is found that a fit can be obtained with two 
shells, and in the case of chlorine with three shells. The 



Distribution of Electrons in Na and CI Atoms. 443 

numbers of electrons on each shell, and the radii of the 
shells, are as follows : — 

Sodium, 

7 electrons on a shell of radius 0'29 A. 
o ,, ,, j, U'/o ,, 

Chlorine. 
10 electrons on a shell of radius 0*25 A. 
5 ., . ., „ 0*86 „. 

3 ., „ „ 1-46 ., 

Secondly, we have solved the simultaneous equations for 
the distribution in shells with several sets of radii, and 
drawn a smooth curve through the points so obtained in 
such a way as to represent the density of distribution of the 
electrons as a continuous function of the distance from 



Fig. 3. 



1 












\ 










I 


1 \ 






















/ 




















\ 


\ 



02 0-4 0-6 0-8 1-0 

Distance from centre of atom in Angstrorr 



the atomic centre. The density P is so defined thatgPdr 
is the number of electrons whose distance from the centre 
lies between r and r+dr. The curves which we obtain for 
sodium and for chlorine are shown in figs. 3 and 4. The 



444 Prof. W. L. Bragg and Messrs, James and Bosanquet : 

total number of electrons in the atom is represented by the 
area included between the curves and the axis. 

Fig. 4. 














































































i 








. 











Distance from centre of atom in Angstrom units. 

The following table shows the agreement between the 
F curves found experimentally and those calculated from 
the electron distributions : — 



Table I. — Sodium. 

Sin 6. 01. 0-2. 0-3. 



0-4. 



0-5. 



^Observed 8-32 


540 


3-37 


2-02 


076 


pj Shells (0;29A| g . 56 


5-59 


333 


219 


0-93 


1 

^ Smooth Curve ... 8*S7 


5-40 


3-20 


1-91 


1-00 



Table II. — Chlorine. 



Sin0. 



0-1. 



0-2. 



0- 



0-4. 



05. 



Observed 12*72 



7-85 


5-79 


4-40 


3-16 


773 


5-90 


4-61 


2-69 


7-80 


5-55 


410 


3-20 



f 0-25 A \ 
F { Shells \ 086 V 12-53 
| I T46 j 

^Smooth Curve ... 12-70 



6. We have also made an approximate calculation ©f the 
F curve to be expected from an atom of the type pro- 
posed by Bohr *, In the ionized sodium atom containing 

* Nature, cvii. p. 104 (1921). 



Distribution of Electrons in Na and CI Atoms. 445 

10 electrons, two are supposed to describe circular one- 
quantum orbits about the nucleus, while, of the remaining* 
eight, four describe two-quantum circular orbits and four 
two-quantum elliptical orbits. We have calculated the size 
of these orbits from the quantum relation ship and the 
charges ; this can only be done very approximately, owing 
to the impossibility of allowing for the interaction of the 
electrons. We take the following numbers : — 

Radius of 1 quantum ring 0*05 A. 

2 „ 0-34 „ 

Semi-major axis of ellipses * ... 0*42 ,, 

To get a rough idea of the diffracting power of such an 
atom, we suppose, first, that the orientation of the orbits is 
random so that the average atom has a spherical symmetry, 
and also that the periods of the electrons in their orbits are 
so large compared with the period of the X-rays that we 
need not consider the effect of their movements. 

The calculation of the effect of the circular orbits offers 
no difficulties. To allow for the effect of ihe ellipses, the- 
following method was used. The elliptical orbit was divided 
into four segments, through each of which the electron would 
travel in equal times. It was then assumed that, on the 
average, one of the four electrons describing ellipses would 
be in the middle of one of these segments. This gives four 
different values of the radius vector, corresponding in the 
average atom to four spherical shells of these radii. 

AYe thus calculate the value of F for an atom having 

2 electrons on a shell of radius 0*05 A. IT. 

4 .. „ „ 0-34 ., 

1 „ „ „ 0-27 „ 

1 .. „ „ 0-55 „ 

1 „ „ „ 0-70 „ 

1 n .<, » 0*78 ,, 



* The elliptical two-quantum orbit of a single electron about the 
sodium nucleus would have a semi-major axis equal to the radius of 
the two-quantum circle. We have used the larger value 0-42 to make 
some allowance for the fact that part of the orbit lies outside the inner 
electrons, so that the effective nuclear charge is reduced. 



446 Prof. W. L. Bragg and Messrs. James and Bosanquet : 
This gives the following figure for F^ a : — 

Sin 9. 0-1. 0-2. 0-3. 0-4. 0'5. 



F calculated 
F observed . . 



•73 


5-04 


376 


.2-53 


1-80 


•32 


5-40 


3-37 


2-02 


0-76 



The agreement, of course, is not perfect, but one must 
remember that no attempt has been made to adjust the size 
■of the orbits to fit the curve. The method of calculation 
too is very rough, although it must give results of the right 
order. The point to be noticed is that the curve is quite of 
the right type, and there is no doubt that an average distri- 
bution of electrons of the nature given by such an atom 
model could be made to fit the observed value of F quite 
satisfactorily. 

7. The points which appear to us to be most doubtful in 
the above analysis of our results are the following : — 

(a) We have assumed that each electron scatters inde- 
pendently, and that the amount of scattered radiation is that 
calculated for a free electron in space according to the 
classical electromagnetic theory. It is known that for very 
short waves this cannot be so, since the absorption of 7 rays 
by matter is much smaller than scattering would account 
for, if it took place according to this law. On the other 
hand, the evidence points towards the truth of the classical 
formula in the region of wave-lengths we have used 
(0-615 A). 

(b) We have used certain formulae (given in our previous 
papers, to which reference has been made) in order to 
calculate the quantity we have called Q in equation (2) 
from the observed intensity of reflexion of a large crystal. 
Darwin * has recently discussed the validity of these 
formulae. The difficulty lies entirely in the allowance which 
has to be made for "extinction" in the crystal. X-rays 
passing through at the angle for reflexion suffer an increased 
absorption owing to loss of energy by reflexion. 

Darwin has shown that this extinction is of two kinds, 
which he has called primary and secondary. If the crys- 
talline mass is camposed of a nuaiber of nearly- parallel 
homogeneous crystals, each so small that absorption in it 
is inappreciable even at the reflecting angle, then secondary 
extinction alone takes place. At the reflecting angle the 

* Phil. Mag. vol! xliii. p. 800, May 1922. 



Distribution of Electrons in Na and CI Atoms. 447 

X-rays suffer an increased absorption, because a certain 
fraction of the particles are so set as to reflect them and 
divert their energy. We made allowance for this type of 
extinction in our work, and Darwin concludes that our 
method of allowance, while not rigorously accurate mathe- 
matically, was sufficiently so for practical purposes. 

Primary extinction arises in another way. The homo- 
geneous crystals may be so large that, when set at the 
reflecting angle, extinction in each crystal element shelters 
the lower layers of that element from the X-rays. Darwin 
has calculated that this will take place to an appreciable 
extent for the (100) reflexion if the homogeneous element 
is more than a few thousand planes in depth. A large 
homogeneous element such as this does not produce an 
effect proportional to its volume, since its lower layers are 
ineffective, and a crystal composed of such elements would 
give too weak a reflexion. Our method of allowing for 
extinction will not obviate this effect. 

We cannot be sure, therefore, that we have obtained a 
true measure of Q for the strong reflexions. The F curve 
may be too low at small angles. It is just here that its form 
is of the highest importance in making deductions as to 
atomic structure. Until this important question of the size 
of the homogeneous elements has been settled, we must 
regard our results as provisional. 

(c) The allowance for the thermal agitation of the atom 
(the Debye factor) is only approximate ; it depends on a few 
measurements made by W. H. Bragg in 1914. In order to 
see how much error is caused by our lack of knowledge of 
the Debye factor, we have calculated the electron distribution 
without making any allowance for it. The result may appear 
at first rather surprising ; the electron distribution so calcu- 
lated is almost indistinguishable from that which we found 
before, when allowance for the Debye factor had been made. 
This is so, although the factor is very appreciable for the 
higher orders of spectra, reducing them at ordinary tem- 
peratures to less than half the theoretical value at absolute 
zero. The difference which the factor makes can best be 
shown by comparing the radii of the shells which give the 
best fit with (1) the F curve deduced directly from the expe- 
rimental results, (2) the F curve to which the Debye factor 
has been applied. 



448 Prof. W. L. Bragg and Messrs. James and Bosanquet : 





(1). 


(2). 




Radius 

(without allowance 
for thermal 
agitation). 


Radius 

(with allowance 

for thermal 

agitation). 


odium. — Seven electrons . 


0-31 


0-29 


Three electrons . . . 


0-79 


0-76 


hlorine. — Ten electrons . . . 


0-28 


0-25 


Seven electrons... 


0-81 


0-86 


Three electrons.. 


1-46 


1-46 



A little consideration shows the reason for this. The form 
of the F curve at large angles is almost entirely decided by 
the arrangement of the electrons near the centre of the atom. 
A slight expansion of the grouping in this region causes a 
large falling off in the intensity of reflexion. This is shown 
in the analysis by the slight increase (O02 to O03 A) in the 
radius of the shell which gives the best fit to the uncorrected 
curve. '1 he effect of the thermal agitation is to make the 
electron distribution appear more widely diffused ; however, 
the average displacement of the atom from the reflecting 
plane owing to its thermal movements is only two or three 
hundredths of an Angstrom unit at ordinary temperatures, 
and so we get very little alteration in our estimate of the 
electron distribution. The uncertainty as to the Debye 
factor, therefore, does not introduce any appreciable error in 
our analysis of electron distribution. 

8. It is interesting to see whether any evidence can be 
obtained as to whether a valency electron has been trans- 
ferred from one atom to the other or not. This may be put 
in another way : can we tell from the form of the F curves 
in fig. 1 whether their maxima are at 10 and 18 or at 
11 and 17 respectively? It appears impossible to do this ; 
and, when we come to consider the problem more closely, 
it seems that crystal analysis must be pushed to a far greater 
degree of refinement before it can settle the point. If all 
the electrons were grouped close to the atomic centres, and 
if the transference of an electron meant that one electron 
passed from the Na group to the CI group, then a solution 
along the lines of that attempted by Debye and Scherrer 
for LiF might be possible. The electron distributions we 
find extend, on the other hand, right through the volume 
of the crystal. The distance between Na and CI centres is 
2*81 A, and we find electron distributions 1 A from the centre 
in sodium and 1*8 A from the centre in chlorine. If the 



Distribution of Electrons in Na and CI Atoms. 449 

valency electron is transferred from the outer region of one 
atom to that of the other, it will still be in the region between 
the two atoms for the greater part of the time, since each 
atom touches six neighbours, and the difference in the 
diffraction effects will be exceedingly small. It is for this 
reason that we think Debye and Scnerrer's results for LiF, 
which were not absolute measurements such as the above, 
were not adequate to decide whether the transference of a 
valency electron has taken place. 

We have assumed that the atoms are ionized in calculating 
our distribution curves. If, on the other hand, we had 
assigned 11 electrons to sodium and 17 to chlorine, we 
should have obtained curves of much the same shape but, 
with an additional electron in the outermost shells of sodium 
and one less in those of chlorine. 

9. Summary. — We have attempted to analyse the distri- 
bution of electrons in the atoms of sodium and chlorine by 
means of our experiments on the diffraction of X-rays by 
these atoms. The results of the analysis are shown in 
figs. 3 and 4. 

The principal source of er.ror in our conclusions appears to 
be our ignorance as to the part played by "extinction" in 
affecting the intensity of X-ray spectra. The distributions 
of the electrons are deduced from the F curves (fig. 1). 
The most important parts of these curves are the initial 
regions at small angles, for errors made in absolute values 
in this region alter very considerably the deductions as to 
electron distribution. The exact form of the curve at large 
angles is of much less interest. Now, it is in this initial 
region, corresponding to strong reflexions such as (100) r 
(110), (222), that extinction is so uncertain a factor. Until 
the question of extinction is satisfactorily dealt with, the 
results cannot be regarded as soundly established. 

If our results are even approximately correct, they prove 
an important point. There cannot be, either in sodium or 
chlorine, an outer " shell " containing a group of eight 
electrons, or eight electrons describing orbits lying on an 
outer sphere. Such an arrangement would give a diffraction 
curve which could not be reconciled with the experimental 
results. Eight electrons revolving in circular orbits of the 
same radius would give the same diffraction curve as eight 
electrons on a spherical shell, and are equally inadmissible. 
On the other hand, it does seem possible that a combination 
of circular and elliptical orbits will give F curves agreeing 
with the observations. 

Phil Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 G 



[ 450 ] 



XLIV. On the Partition of Energy. By C. Gr. Darwin, M.A., 
L.M.S., Fellow and Lecturer in Christ's College, Camb., 
and R. H. Fowler, M.A., Fellow and Lecturer in Trinity 
College, Camb.* 

§ 1. Introduction. 

AN important branch of atomic theory is the study of 
the way in which energy is partitioned among an 
assembly of a large number of systems — molecules, Planck 
vibrators, etc. This study is based on the use of the principles 
of probability which show that one type of arrangement is 
much more common than any other. The most usual method 
is to obtain an expression for the probability of any state 
described statistically and then to make this probability a 
maximum. This always involves a use of Stirling's approxi- 
mation for factorials, which in many cases is illegitimate at 
first sight, and though it is possible to justify it subsequently, 
this justification is quite troublesome. It is also usually 
required to find the relation of the partition to the temperature 
rather than to the total energy of the assembly, and this is 
done by means of Boltzmann's theorem relating entropy to 
probability- — a process entailing the same unjustified approxi- 
mations. 

The object of the present paper is to show that these 
calculations can all be much simplified by examining the 
average state of the assembly instead of its most probable 
state. The two are actually the same, but whereas the most 
probable state is only found by the use of Stirling's formula, 
the average state can be found rigorously by the help of the 
multinomial theorem, together with a certain not very 
difficult theorem in the theory of the complex variable. By 
this process it is possible to evaluate the average energy of 
any group in the assembly, and hence to deduce the relation 
of the partition to temperature, without the intermediary of 
entropy. The temperature here is measured on a special 
scale, which can be most simply related to the absolute scale 
by the use of the theorem of equipartition, and we shall also 
establish the same relationship directly by connecting it with 
the scale of a gas thermometer. Throughout the paper the 
analysis is presented with some attempt at rigour, but it will 
be found that apart from this rigour it is exceedingly easy to 
apply the method of calculation. Most of the results are not 

* Communicated by the Authors. 



On the Partition of Eneryy. 451 

new ; it is the point of view and the method which, we 
think, differ from previous treatments 

No discussion of the question of' partition would be com- 
plete without consideration of its relation to thermodynamic 
principles. We shall leave this view of the subject to a 
future paper ; for the iucreased light thrown on the statistical 
nature of entropy raises many interesting points which could 
not be discussed here properly without making the present 
work run to inordinate length. 

§ 2. Statistical Principles and Weiyht. 

Before proceeding to the problem it will be well to review, 
in general outline, the principles of the theory of the 
partition of energy, though we have nothing new to say in 
this connexiou. We shall be concerned with collections of 
molecules, Planck vibrators, etc. — each individual unit will 
be called a system, and we shall call the whole collection an 
■assembly. We shall be dealing mainly with assemblies com- 
posed of groups of systems, the individuals in each group 
being identical in nature. In order to make the problem 
definite it is necessary to assume that each system has 
a definite assignable energy, and yet can interact with the 
others. This requires that the time of interactions, during 
which there will be energy which cannot be assigned to a 
definite single system, is negligibly small compared with 
the time during which each system describes its own 
motion. 

For such an assembly we are to calculate various average 
properties of its state, when it describes its natural motion 
according to whatever laws it may obey. There will at any 
rate be an energy integral, and we have therefore to calcu- 
late these averages subject to the condition of constant 
energy. To determine the basis on which these averages 
are to be calculated we are to apply the principles of proba- 
bility ; and the calculation of itself falls into two stages, the 
prior and the statistical. The prior stage aims at establishing 
what are the states which are to be taken as of equal 
probability. In the statistical stage we have simply to 
enumerate the states specified in the prior stage, allow for 
the fact that the systems are macroscopically indistin- 
guishable, and evaluate the averages taken over these 
states. 

It is not here our purpose to enter into a full discussion 
of the fundamental questions that arise in connexion with 
the determination of what states ought to be taken as equally 

2 G2 



452 Messrs. C. Gr. Darwin and R. H. Fowler on 

probable. It will suffice to recall that for assemblies obeying 
the laws o£ classical mechanics the theorem of Liouville 
shows that the elements of equal probability may be taken 
to be equal elements of volume in Gibbs' " phase space." 
It follows out of this, for example, in the case of an assembly 
of a number of identical systems — say simple free mole- 
cules — that the elements of equal probability can be simpli- 
fied down into 6-dimensional cells dq l dq 2 dq z dp l dp 2 dp z of 
equal extension, where q l9 q 2 , q 3 are the coordinates, and 
Pi> Pi, ps the conjugated momenta, of a single molecule. 
We shall describe this by saying that the weight of every 
equal element dq A ... dp z is the same, and by a slight 
generalization, that the weights of unequal cells are pro- 
portional to their 6-dimensional extension. The word weight 
is here used in exactly the sense of the term a priori 
probability, as used by Bohr and others. 

But when we come to the quantum theory, mechanical 
principles cease to hold, and we require a new basis for 
assigning the equally probable elements. Such a basis is 
provided by Ehrenfest' s * Adiabatic Hypothesis and Bohr's f 
Correspondence Principle. These show how the theorem of 
Lionville is to be extended, and allow us to assign a weight 
for each quantized state of a system. It is found that we 
must assign an equal weight to every permissible state in 
each quantized degree of freedom. At first sight this is a 
little surprising, for it would seem natural to suppose that a 
vibrator which could only take energy in large units would 
be less likely to have a unit than one which could take it in 
small ; but this is to confuse the two stages of the problem. 
It is only by the supposition of equal weights that we can 
obtain consistency with classical mechanics by the Corre- 
spondence Principle. It is customary % in assigning a definite 
weight to every quantized state to give it the value h, so as to 
bring the result to the same dimensions as those of the 
element dq dp in the classical case. But there is considerable 
advantage in reversing this, and taking the quantized weights 
as unity and the weight of the element in the phase space as 
dq dp/h ; for if this is done, the arguments about entropy are 
simplified by the absence of logarithms of dimensional 
quantities. We shall adopt this convention here, though in 

* Ehrenfest, Proc. Acad. Amst. xvi. p. 591 ; Phil. Maof. xxxiii. p. 500 
(1917), etc. 

t Bohr, " The Quantum Theory of Line Spectra," Dan. Acad. iv. p. 1 
(1918). 

% Ehrenfest & Trkal, Proc. Amst. Acad. Sc. xxiii. p. 162. See in par- 
ticular p. 165 and Additional Notes, No. 1. 



the Partition of Energy. 453 

all our results it is immaterial — indeed, until such questions 
as dissociation are considered it makes no difference to adopt 
different conventions for different types of system. The 
convention has the advantage of shortening a oood manv 
formula? and freeing them from factors which are without 
effect on the final results. 

An exception to the above rule for assigning- weights to 
quantized motions occurs in the case of degenerate si/stems, 
where there are two degrees of freedom possessing the same 
or commensurable frequencies. In this case there is only 
one quantum number, and the state of the system is partly 
arbitrary. Bohr * shows that the rational generalization is 
to assign to such a state a weight factor which can be 
evaluated by treating the system as the limit of a non- 
degenerate system, and quantizing it according to any pair 
of variables in which it is possible to do so. The number of 
the permissible states which possess the same total quantum 
number will give the weight of the state. A corresponding- 
rule holds for systems degenerate in three or more degrees 
of freedom. 

The meaning of weight can perhaps be made clearer by 
considering its introduction the other way round — beginning 
with an assembly of simple quantized systems of various 
frequencies. Griven the energy, there is a definite number 
of possible states, which are fully specified by the energy 
assigned to each system. We then make the hypothesis that 
it is right to assign an equal probability to each such state 
in the calculation of averages. This is now the fundamental 
postulate. The generalization to degenerate systems goes as 
before, by introducing weight factors. Finally, passing 
over to mechanical systems, such as free molecules, we are 
led by an appeal to the converse of the Correspondence 
Principle to attach weight dq x . . . dp^/h 3 to each 6-dimen- 
sional cell which specifies completely the state of a single 
molecule. 

The second, statistical, half of the problem consists in 
enumerating the various complexions possible to the assembly. 
By a complexion we mean every arrangement of the assembly, 
in which we are supposed to be able to distinguish the in- 
dividuality of the separate systems. We count up the total 
number of complexions which conform to any specified 
statistical state of the assembly, and attach to each the 
appropriate weight factor. Thus the probability of this state 
is the ratio of the number of its weighted complexions to the 

* Bohr, loc. cit. p. 26. 



454 Messrs. C. G. Darwin and R. H. Fowler on 

total number of all possible weighted complexions. This 
part of the problem depends on the nature of the particular 
assembly considered, and so must be treated separately in 
each case. 

We start in § 3 with a problem which concerns not the 
partition of energy, but the distribution of molecules in a 
volume. It illustrates the method in its simplest aspect and 
has the advantage of being purely algebraic. Next, in § 4, 
we take the distribution of energy among a set of similar 
Planck vibrators, which is again a purely algebraic process, 
and then proceed in § 5 to introduce the main theme of this 
paper by dealing with the partition of energy between two 
sets of Planck vibrators of different type. This is most 
conveniently treated by using the complex variable, and in 
§ 6 there is a discussion of the required theorem. The par- 
tition of itself introduces the temperature, and in § 7 the 
special scale is compared with the absolute. In §§ 9, 10, 11 
the partition law is generalized to more complicated types of 
system, such as the quantized rotations of molecules. In 
§§ 12, 13 the method is extended so as to deal with the free 
motion of monatomic molecules, intermixed with vibrators. 
The work leads to a rather neat method of establishing the 
Maxwell distribution law. 

§ 3. The Distribution of Molecules in Space. 

The first example we shall take is not one of a partition of 
energy, but of the distribution of small molecules in a vessel. 
It illustrates in its simplest form the averaging process, 
and has the advantage of depending only on elementary 
algebra. 

Let there be M molecules, and divide the vessel into m 
cells of equal or unequal volumes r l5 v 2 ... v m , which may 
each be as large or as small as we like. Then 

Vi + V 2 + ... +v m = V (3-1) 

By well-known arguments which we need not consider, it 
follows that any one molecule is as likely to be in any element 
of volume as in any other equal one. So by a slight ex- 
tension of the idea of weight we attach weights v h v 2 , ... v m to 
the cells. To specify the statistical state we say that the 
first cells has a x molecules, the second a 2 , and so on. Then 

a 1 + a 2 -\- ... +a m = M (3*2) 

By the theory of permutations the number of complexions 



the Partition of Energy, 455 

which conform to the specification is 

M! 

a x ! a 2 ! . . . a m I ' 

and each of these must be weighted with a factor 

u l u % • • • ' »i 

The total number of all the weighted complexions is 
ai ! a 2 ! . . . 

= (, 1+! . 2+ ...) M = VM, 

by the multinominal theorem. This could have been deduced 
at once by working direct with probabilities v r /Y instead of 
weights v y , but the argument has been given in detail to 
illustrate the method for more complicated cases. 

We next find the average value of a r . This is given by 

M' 
Qa r = % a a r , vfivf* ... . 

Ui 1 «2 ' • • • 

To sum this expression we only have to cancel a r with the 
first factor in a r ! in the denominator, and then it is seen to 
be equal to 

Mv r (v l + v 2 + ...yi-\ 

and so, as is implicit in our assumptions, 

a, = Mt V /V (3'3) 

But we can now go further and find the range over which 
a r will be likely to fluctuate. This is estimated by averaging 
the square of the difference of a r from its mean value. We 
shall throughout this paper describe such a mean square 
departure as the fluctuation of the corresponding quantity. 
Thus the fluctuation of a r is (ar — a r ) 2 . Now 

(a r — a r ) 2 = a r (a r — 1) + a r — 2a ~d r -\-~a r 2 , 

and averaging the separate terms by the multinominal 
theorem, we have 

(a r a r ) — y 2 + y V ' V + \ V / 

= ^(l-v) = -( 1 -M> • • • ^ 

This result represents the fluctuation however large or small 



456 Messrs. C. G. Darwin and R. H. Fowler on 



v r may be. In all cases we have the result that (a r — a r ) 2 is 
less than a~ r , and therefore that the average departure of 
a r from ~a r is of order (o^)*. We can also interpret this fact 
by saying that departures of a r from li r which are much 
greater than (a7-)* will be relatively rare ; as M is large and 
(or)i small compared with oT r , this is precisely equivalent to 
saying that the possession of the average value of a r is a 
normal property of the assembly in the sense used by Jeans *« 
We have thus a simple and complete proof that uniform 
density is a normal property of this assembly. 

§ 4. The Distribution of Energy among a Set of 
Planck Vibrators. 

Another case where the treatment can be almost entirely 
algebraic is that of the partition of energy among a set of 
Planck vibrators which all have the same frequency. Let e 
be the unit of energy so that every vibrator can have any 
multiple of e. As we saw in § 2, the weight attached to every 
state is to be taken as unity. 

Let there be M vibrators and let there be Pe of energy 
(P is an integer) to be partitioned among them. To specify 
a statistical state, let a be the number of vibrators with no 
energy, a x with e, a 2 with 2e, etc. Then we have 

a + a 1 + a 2 + a 3 + ... = 'M., .... (4'1) 

a 1 + 2a 2 + 3a 3 +... = P, .... (4'2) 

and anv set of a's which satisfy these equations corresponds 
to a possible state of the assembly. By the principles of § 2 
each of the complexions will have unit weight. Now count 
up the number of complexions corresponding to the speci- 
fication. By considering the various permutations of the 
vibrators, it is seen to be 

M! 



a \ ail a 2 l 



(4-3) 



We must next find C the total number of all possible 
complexions. Let X a denote summation over all possible 
values of the a's which satisfy (4*1) and (4*2). Then 

a ! aj : a 2 l ... 
Consider the infinite series 

(i+s + si +5 3 + ; >)M - 

* Jeans, ' Dynamical Theory of Gases/ passim. 



the Partition of Energy. 457 

■expanded by the multinominal theorem. The typical term is 

a ! a x ! a 2 \ ...~ 

where the as take any values consistent with (4*1). Then 
if we pick out the coefficient of z F , we have the sum of all the 
expressions for which the a's satisfy both (4*1) and (4*2). 
Observe that we may take the whole infinite series because 
the later terms are automatically excluded. 

Now this will be the coefficient of X s in (1—z) ~ M , and so 

(M + P-l)! a . 4 . 

°- (M_i)!P! ' ^** J 

which is the ordinary expression for the number of homo- 
geneous products as formerly used by Planck *. 
We next evaluate the average of a r ; 

M! 

a ! a l ! a 2 1 . . . 

-MS, ( M -^ ! 

a ! a l ! a 2 I . . . 

where X a > denotes summation over all values satisfying 
«o' + «i / + « 2 / + «3 / --- =M — 1, 

a/ + 2a 2 / 4 3a 3 / ... = P — r. 
The sum is thus 

M (MtP-r-2)! 
(M-2)!(P-r)!' 
and we have 

q-M(M n ( M + P-r-2)! P! 

This is exact, and holds for all values of r ; now r can have 
any value up to P and the majority of the a r 's will be zero. 
The ordinary method of proof applies Stirling's formula for 
a r ! to these zero values. In the important case where both 
M and P are large, it will be only necessary to consider 
values of r which are small compared with P. Now, if r 2 is 
small compared with P, P!/(P — r)! has the asymptotic 
value P r . Using relations of this type and also disregarding 
the difference between M and M — 1, we have 

M 2 P r 

"*= (M+ty+i (4 ' 5) 

* M. Planck in the earlier editions of his book on Radiation. 



458 Messrs. C. Gr. Darwin and R. H. Fowler on 

The same methods give the fluctuations of a r . For 



(a r — a r ) 2 — a r a r — 1) -f a r — (a r ) 2 , 
and a process similar to the above gives 

0-^?^=T) = M(M - X) 

The exact expression for the fluctuation can be at once put 
down. When M and P are taken large the leading term 
cuts out, and so it is necessary to carr j the approximation to 
the next order. If we substitute 

P!/(P-r)I--P^-ir(r-I)F- 1 , 

we find that 



\(Xf (Jby J — 

£ { l ~ (M + Pr* [2P 2 -(2?-l)MP + r*W] j . (4-6) 

The formula for a r can be put into a more familiar form by 
the substitution P = M/(^ a — 1), which gives 

~a r = Me- ra (l — e- a ), . . . . (4 ! 7) 

and leads to a corresponding but more complicated ex- 
pression for the fluctuation. Here, as we shall see later, ol 
can be identified with the familiar e/kT., Equation (4* 6) 
establishes at once that the statistical state specified by (4*5) 
or (4* 7) is a normal property of the assembly. 

§ 5. The Partition of Energy among two Sets 
of Planck Vibrators. 

After these preliminary examples we now apply our method 
to a problem which will bring out its distinctive character, 
that of the partition of energy in an assembly composed of 
different types of system. We shall consider first the simplest 
of such cases — an assembly consisting of a large number of 
Planck vibrators of two types A and B. The number of A's- 
is M, and the energy unit of an A is e as before. There are 
now also N B's with energy unit tj. To make exchanges of 
energy possible we have to suppose, say, that there are 
present a few gas molecules, but that the latter never possess 
any sensible amount of energy. (Later on in § 12 we develop 
a method by which we shall be able to include any number 
of such molecules in our assembly.) We also require for 
the purposes of the proof to assume that e and rj are com- 
mensurable, but it does not matter how large the numbers 
"may be which are required to express the ratio e/rj in its 



the Partition of Energy. 459 

lowest terms. To avoid introducing new symbols, we may 
suppose that the unit of energy is so chosen that e and r\ are 
themselves integers without a common factor. 

We have already introduced the idea of weight, and seen 
that we must assign the weight unity to every permissible 
state of a linear vibrator. To calculate the number of com- 
plexions of the assembly of any given sort, we have merely 
to calculate the number of ways in which the energy may 
be distributed among the vibrators, subject to the given 
statistical specification. A simple example will make the 
process clear. 

Let there be two A's and two B's; let ^ = 2e, E = 4e. 
Then the possible complexions are : — 



{aaaa 
r i r 
a a a < 

( aaaa' 
I aa'a'a 



aab 
aab' 
a'ab 
Wa'b' 



job 



\aa!b 

\aa'V bb' 



Here, for example, aaaa' means that there is 3e of energy 
on the first of the A's, e on the second, and none on the B's. 
Each of the fourteen complexions is, by definition, of equal 
weight, and is therefore to be reckoned as of equal probability 
in the calculation of averages. Observe how with the small 
amount of energy available a, good deal more goes into the 
smaller than into the larger quanta ; for the pair of A's have 
on the average ye, as against ye for the pair of B's. 

We pass to the general case. The statistical state of the 
assembly is specified by sets of numbers a r , b s where a r is the 
number of vibrators of type A which have energy re, and b s 
the number of B's with energy stj. All weights are unity 
and the number of complexions representing this statistical 
state is the number of indistinguishable ways (combinations) 
in which M vibrators can be divided into sets of a , a Y ... and 
at the same time N into sets b , b ± ... . As illustrated by 
the example, it is therefore given by the formula 

M! N! 

o !a l la 1 !...* a !6 l !W..." * * * ( ; 

In (5"1) a r and b s may have any zero or positive values 
consistent with the conditions 

2 r a r = M, SA = N, 2 r r6a r + X s s V b s = E, . (5'2) 

where E is the total energy of the system — necessarily an 



460 Messrs. C. G. Darwin and It. H. Fowler on 

integer in the units we employ. The total number C of all 
complexions is therefore 

M ! N ! 

G = ^ b aJa'JaTj::. b l b ± l b 2 \ ... ' * * (5 ' 3) 

where the summation X a ,b is to be carried out over all positive 
or zero values of a r and b s which satisfy (5*2). 

By using (5*1) and (5*3) we can at once obtain an expression 
for the average value, taken over all complexions, of any 
quantity in which we are interested. We have already 
studied ~a T in § 4. The main interest centres in Ea, the 
average energy on the A's. We have at once 

PF -? (2,.r £ q r )M!N! 

^-^A — A*, b i 1 1 7 I 7 I 7 I • • • \p ^) 

The following process leads to simple integrals to express 
the quantities C, CEa, etc. Consider the infinite series 

(l + z e + z 2e + ) M 

expanded in powers of z by the multinominal theorem. The 
general term is 

'Ml z ? r rea„ m 

a Q ! % ! ..." 

It follows that if we select from the expansion of 

(l + * e + ^ e +...) M (l + s*^* + ...)* . (5-5) 

the coefficient of z E , we shall obtain the sum of all possible 
terms such as (5*1) subject to the conditions (5*2), that is to 
say C Similarly, if we form the expression 

j^(l+s e H-2 2e ...) M 1(1-^ + ^...)*, . (5-6) 
the general term in the first bracket must be 

{% r T€a r )M\ 2 r rea r 
— Z , 






\^\ax\ 



and by the same reasoning the coefficient of z E in (5*6) must 

be CE A . 

Expressions (5*5) and (5*6) are easily simplified— they are 
respectively 



the Partition of Energy. 461 

The latter can also be written as 



{-Mjrgglog (!-<«) }(l-*«)-*(l. 



If these expressions are now expanded in powers of z by 
the binomial theorem, they give a sum of products of factorials 
which are, of course, the " homogeneous product " expressions 
used by Planck. It is possible to approximate to these by a 
legitimate use of Stirling's theorem and to replace the sums 
by integrals without much difficulty. It would, indeed, have 
been possibly to start from these expressions, but we have 
not done so because in the general case to be discussed later 
that method would not be available. To make further 
progress* by a method of general utility, we discard Stirling's 
theorem and express these coefficients of z® by contour 
integrals taken round a circle y with centre at the origin and 
radius less than unity. By well-known theorems on in- 
tegration * we have at once 



c 



i r dz i . 

= 2mUz E+1 (l-z e ) M (l--z r i)K> ' ' (5 ' 71) 



d 



-Ms-T-logd-s 6 ) 



— 1 C d~ ~ dz & K J 

CEA= 2^-J y ^i-(1_^M (1 _^ '• ' ( 5 * 72 > 

AVe can no longer hope for the single-term formulae of 
§§ 3 ; 4. But (5-71) (5-72) are exact, and when M, N, E are 
all large in any definite fixed ratios, we can make use of the 
method of steepest descents to obtain simple adequate approxi- 
mations. The method is very powerful and can be applied 
in a great number of cases without difficulty. Moreover, it 
is comparatively easy to use it with mathematical rigour if 
that is desired ; and thus the somewhat clumsy calculations 
in the usual proofs of partition theorems are entirely avoided. 

In general terms the process is this. Consider the in- 
tegrand on the real positive axis. It becomes infinite at z = 
and again at z=l, and somewhere between at z = § there is a 
minimum which is easily shown to be unique. Take as the 
contour the circle with centre at the origin and radius S 
passing through this minimum. Then we find that, for 

* For those not familiar with these theorems we may remark that 

5— -. 1 2 r <7z=0 when r is any integer other than — 1, while r> — ' 1 — =1*;. 

these equations at once give (5'71) and (5 - 72). 



462 Messrs. C. G. Darwin and R. H. Fowler 



on 



values of z on the contour, z = § corresponds to a strong maxi- 
mum, and when M, N, E are large, such a strong maximum 
that practically the whole value of the integral is contributed 
by the contour in the neighbourhood of this point. Hence 
in the integrals it is legitimate to substitute the value at this 
point tor any factors which do not themselves show strong 
maxima here or, elsewhere. On this general principle we 

can remove the term — Wz -j-^ log (1— z e ) from under the 

integral sign, provided that z is given the value S determined 
by the maximum condition. The part of the integrand in- 
volving the large numbers M, N, E, determines the value of 
■■& as being the unique real positive fractional root of the 
equation 

j- |^- E (1-0 6 )" M (1-^)" N }=O. 
That is, 3 satisfies the equation : 

The remaining integrands in C and CEa are identical, and we 
therefore have 

El=-M^log(l-n 

-A n 

If a similar process is carried out for the B's, we have 

s^plrr ( 5-91 ) 

in agreement with the necessary relation 

Ea + Eb = E. 

Equations (5*9) (5*91) determine the partition of energy and 
take their familiar form if we replace S by e~ 1/kT . We shall 
return to this point later. 

§ 6. Application of the Method of Steepest Descents. 

After this sketch it will be well to establish the validity of 
the ar uments used. This section is put in for mathematical 
completeness, and is not concerned at all with physical 
questions. We treat of a more general case than that of § 5. 



the Partition of Energy. 463 

Arguments of tins type — asymptotic expansions by steepest 
descents — are, of course, well known in pure mathematics. 
Consider a contour integral of the form 

isj/WW-)]"* .... (6-1) 

subject to the following conditions : — 

(i.) <\>{z) is an analytic function of z, which can be ex- 
panded in a series of ascending powers of z. 

(ii.) This series starts with some negative power. 

(iii.) Every coefficient is real and positive. 

(iv.) Its circle of convergence is of radius unity. (This 
condition is quite unessential to the mathematics, 
but makes the statement simpler, and is physically 
true.) 

(v.) The powers that occur in the series cannot all be put 
in the form a + /3r- where a and ft are any given 
integers and r takes all integer values. 

(vi.) F(r) is an analytic function with no poles in the unit 
circle, except perhaps at the origin, 
(vii.) 7 is a closed contour going once counterclockwise 
round the origin. 

The problem is to obtain the asymptotic value as E tends 
through integer values to infinity. 

We shall first study the properties of <j>(z). Considering 
real values, it must have one and only one minimum between 
and 1. For it is continuous and tends to + co at both 
and 1, and so must have at least one minimum between. 
Further, to find minima we differentiate, and then all the 
negative powers will have negative coefficients and all the 
positive positive. It follows that minima are given where 
two curves cut, one of which decreases steadily between 
and 1, while the other increases steadily. These curves can 
only cut in one point, and so there is only one minimum. 
Xext, for the complex values, consider a circle of any radius 
r less than 1. As the modulus of a sum is never greater 
than the sum of the moduli, it follows that at no point on this 
circle can \<j>(z)\ be greater than <f>(r). Moreover, it can only 
equal 4>(r) provided that condition (v.) is broken, and in that 
case there will be /3 points at each of which \<j>(z)\ = (j>(r). 
We can thus see that on account of the large exponent it is 
only the part of <y near the real axis that contributes effec- 
tively to the asymptotic value of the integral. This suggests 



464 Messrs. 0, Gr. Darwin and R. H. Fowler on 

the substitution z = re iQI , with a as the new variable of integra- 
tion. Expanding near the real axis we have 

[<K*)] E = W>0)] E exp. {iaEr<£7</>+ 0(E« 3 )}. (6'11) 

This function contains a periodic term of high frequency 
which cuts down the contributions for small values of a, so 
that the value of the integral will in general depend on more 
distant parts of the contour than those to which this approxi- 
mation will apply. But if we choose for r the special value 
3 corresponding to the unique minimum <$>' = 0, then the 
oscillating term in the exponential vanishes and the contribu- 
tions for small values of a dominate the whole integral. For 
this special value of r the exponential becomes 

exp.{-Pa 2 d 2 (/)' , /</) + 0(Ea 3 )}, . . (6-12) 

and by (iii.) <j> f/ >0. We see at once that we can suppose 
that Wa ranges effectively over all values from — oo to + oo 
while all other terms, such as a,..., Ea 3 , ... remain small. 
We then obtain for (6*1) on putting z = §e ia the asymptotic 
expression 

— OG^P f {F(S) + ; a 3F'(3) + 6>0 2 ) + O(E« 3 )}6-PaW70^ 

For most purposes the first term in the expansion will 
suffice, but if the precise values of the fluctuations are re- 
quired, the second also is necessary. As it is in general 
rather complicated, we shall content ourselves here with 
pointing out its order of magnitude. 

On carrying through the necessary calculations we find 

The argument of F and <£ is everywhere 3 ; the term 
{§, </>, F} denotes a complicated expression of ■&, <£ and its 
first four derivatives, and F and its first two derivatives, but 
is independent of E. If condition (v.) is dropped, we shall 
have /3 equal maxima arranged round the circle 7, and, pro- 
vided F has the same value at each of them, the integral will 
have a value equal to (6'2) multiplied by J3. 

Now consider the problem of § 5, to which our work applies 
immediately with 

We may suppose that E tends to infinity and also M and N 



the Partition of Energy. 4G5 

in such a way that M/E and N/E are constant. This func- 
tion satisfies all the conditions of this section — the fact that 
it is in general many-valued is irrelevant, for we are only 
concerned with that particular branch which is real when z 
is real and 0<^<1 and this branch is one-valued in the unit 
circle. The unique minimum S is determined by the equa- 
tion cj>'=0 or 

Me N^ 

E =s=zzi + $=v-l ( fi "4) 

The value of the integral (5*71) is then by {6'2) (omitting 
the second term the form of which is only required in 
calculating fluctuations), 

s-B(i-y)-M(i-s*)-g 

If, contrary to hypothesis, we had taken e and rj as having 
a common factor /3, condition (v.) would have been violated. 
In this case C would be /3 times as great as before, but so 
would CE A , so that E A and all other averages would be 
unaffected. The use of (6'2) to evaluate CE A (5 - 72) etc. 
leads at once to expressions similar to (6*5), and so to the 
results given formally in the last section and to others to be 
given later. 

As we shall see, ■& is the temperature measured on a special 
scale, and there is great advantage in regarding 3, rather 
than E, as the independent variable which determines the 
state of the assembty. If this is done the expression 
~E$ 2 (f>"/(f) can be put into simpler form. For it is easily 
verified that 

Eayv<ft= ^* * + v 



(a- e -i) 2 ' (&-"-i) a ' 
=*SIM»>=*§' ■ • ( 6 " 6 ) 

if E is regarded as a function of S, given by (5'S). 

It may be remarked that the constant occurrence of the 
operator $d/d§ suggests a change of variable to log 3. 
Though this has some advantages we have not adopted it, 
partly because it makes the initial argument about the 
multinomial theorem a little harder to follow and complicates 
the contour of the integration, and also because log $ is not 
itself the absolute temperature — if it had been, the physical 
simplicity might have outweighed the other objections — but 
only a quantity proportional to its reciprocal. 

Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 H 



466 Messrs. 0. G. Darwin and R. H. Fowler on 

§ 7. The Meaning of '&. 

Returning to the subject of § 5, we see that the partition 
of energy between the A's and the B's depends on a para- 
meter which can be determined in terms of" the energ}- by 
means of the equation (5*8). It is fairly evident that 3 is 
connected with the temperature, though it requires more 
general considerations to prove this properly. But if we 
assume this connexion and are content to replace the thermo- 
dynamic definition of absolute temperature by one based on 
the law of equipartition of energy for systems obeying the 
laws of classical mechanics, then we can at once identify 
the meaning of $. 

For let us suppose that the B's are vibrators of very low 
frequency. They will then obey the classical laws, and the 
average energy of each will be kT. But 

Tit v — — £T 

^Js-i-i.-iogCi/*)-*- 1 ' 

which shows that 

*=«-^ T (7-1) 

Substituting this in (5*9), we obtain the well-known form 

Me 



E A = 



e e/kT 



Observe that while rj is tending to zero there need be no 
difficulty about the condition that rj and e are to be com- 
mensurable. We shall later return to this question of 
temperature and establish it for much more general types o£ 

system. 

§ 8. Other mean values. 

Exactly the same methods can be applied to evaluate any 
other mean value besides E A . For example : — 

p - _ v jvn ni 

Ua r-A,» a r^j ai !. ... 6 ! b, !....' 

_ (M-l)l N! 

- M ^WUi'!.... MM-.:..:' * ( } 

summed over all zero and positive values such that 

X r a r ' = M-l, 2A=N, % r r*a r ' + 2,sf,b=$-re. 
Applying the multinomial theorem and reducing the 



the Partition of Energy. 467 

expression to a complex integral, we have 

which, by virtue of the value of C from (5' 71) and the 
argument of § 6, at once yields 

a;=Mr(i-y), 

=M*-«/**(l-e-*/* T ) J . . . . (8-3) 

which is the formula of § 4 over again, the presence of the 
B's being immaterial. 

When we come to evaluate fluctuations the matter is a 
little more complicated, because the leading terms cut out, 
and so the second term of the asymptotic expansion will in 
general play a part. For example, consider the fluctuations 
of a_ : 



( a r- a r) 2 = a X a r- 1 ) + ^-<X) 2 ' 

By arguments exactly similar to those above, we have 



C«„( 



a.-l^MCM-D^j^Tw 



(1_/)N- 2 (1_^)*" 

and so by (6-2) 



«>,- 1) =M(M-ip 2 " £ (l-y) 2 -U+ 0(1/E) }. 

Thus the fluctuation is 

^_M3 2r6 (l-3 e ) 2 + 0(M 2 /E). 

This is sufficient to show that the possession of ~a~ r (8'3) is a 
normal property of the assembly. 

The complete calculation of the O-term is rather com- 
plicated ; the result is given at the end of this section. But 
a great simplification arises if we suppose that there are 
many more B's than A's, while E is so adjusted that $ the 
temperature is unchanged. In this case the term 0(M 2 /E) 
becomes small and may be neglected. We shall describe 
this case by saying that A is in a bath of temperature S. 
Then, provided this is so, we have 

(^EJ=a r Q--aJ-M.) (8-4) 

A much more important quantity is the fluctuation of E A . 
This is found by evaluating E A 2 . Now, just as CE A was 
given by operating with zd/dz on the first factor in 

2R2 



468 Messrs. C- G. Darwin and R. H. Fowler on 

(1 — 2 6 )~ M (1— ^) -N , so CE A 2 is easily seen to be given 

by operating with (~^) in the same way. Thus 

ce?= ^[% { («£) Wr- } (i-^)-*. 

If we again suppose an infinite bath of temperature $, we 
can omit the second term of the asymptotic expansion (6'2) 
and obtain 

e>(i_*O m {(^) 2 (i-*T m }, 

=(i-y) M ^{A(i-*r*b 

— <^7 

and so the fluctuation is 



(E A -E A )*=E7-(E A )* 



-iKP ^ 



(8-5) 



J 



This is a result of which Einstein * made use in his work 
on fluctuations of radiation. It should be emphasized that 
these results are only accurate in a temperature bath, and 
not when the number of systems A is a finite fraction of the 
assembly. 

In all cases (6*2) shows that the possession of E A is a 
norma! property of the assembly. 

If we work out exactly the second terms in the asymptotic 
formulse of § 6 and apply them to the fluctuations of a r and 
E A we find 

(a r -a,)'-a r [l- g |l+ ^ E /cft J J' < 8 6 > 

^^-^^^fV^f) • • • <*?> 

Formula (4*6) above is a special case of (8*6). 

* A. Einstein, Phys. Zeitschr. vol. x. p. 185 (1909). 



the Partition of Energy. 469 

Finally it is of some interest to point out that we can 

obtain a formula for (Ea— "Ea) & of general validity. We 
have in fact 



(E A -E A ) 2 * = 1.3...'(2s-l){(E A -E A ) 2 }*, . ($•$) 



where (E A — E A ) 2 is given by (8*7). We retain of course 
only the highest order term *, which is thus 0(E A )*. 

§ 9. Generalization to any number of types of system, and 
to systems of any quantized character. 

It is clear that the present method of treating partitions 
is of a much more general character than has so far been 
exhibited. Consider an assembly composed of two types, 
A and B, of quantized systems more complicated than Planck 
vibrators. We suppose generally that the systems of type A, 
M in number, can take energies to the extents e , e 1} e 2 , ..., 
and these states have weight factors po^Pi->I ) 2^ ...in conformity 
with the discussion in § 2. Similarly, the B's, N in number, 
can take energies rj , 77^ tj 2i ... with weights q , g^ q 2 , 
We have to suppose that it is possible to determine a basal 
unit of energy such that all the e's and n's can be expressed 
as integers. Further, it simplifies the work if we suppose 
that there is no factor common to all of them. Proceeding 
exactly as before, we set down the weighted number of com- 
plexions which correspond to the specification that, of the A's, 
a r have energy e r ; of the B's, b s have energy ij s . This 
number is 

a ,. , wv • • • • b , ; ?«v . . . ., • (9-i) 

a Q i a 1 i . . . o ! o 1 : . . . 

and the a's and 6's are able to take all values consistent with 
\a=U, 2A = N, S r ^+S^A=E. . (9-2) 
Now form the functions 

f(z)=p a z e « + /h ^+p.^ +...., . . . (9-3) 
9 (z)=q z"° + q^+q,^+ (9-31) 

These will be called the partition functions f of the types of 

* Cf. Gibbs' < Statistical Mechanics,' p. 78. But (8*8) is generally 
valid, while Gibbs' formulae really refer only to a group of systems in a 
temperature bath. 

t They are practically the " Zustandsumme '' of Planck, ' Radiation 
Theory/ p. 127. 



470 Messrs. C. G. Darwin and R. H. Fowler on 

system A and B. The application of the multinomial 
theorem then leads to the consideration o£ the expression 

[/(*)] M M*)f, 

and pursuing exactly the same course as in § 5, we find 

C=2-y^ 1 [/«] M [^)] N , • • • (9-4) 

Assume for the moment that we can choose a <\>(z) con- 
forming to the requirements of § 6. The whole calculation 
then goes on as before. The radius of the circle to be taken 
as contour is given by the equation 

E = M^log/(3)+N^ log </(£). . . (9-6) 

This equation has one and only one root. We thus can 
at once put down 

, *> \ . . . (9-7) 

=M3 J log /(d). ) 

In exactly the same way we have 

£=Mp,**/X»). (9-8) 

and we can also verify that in the case of an infinite bath 
the fluctuations are again given by (8*4), (8*5), and that 
equation (6'6) is still true. The exact forms of the fluctua- 
tions {8'$)i (8*7) are also valid if we replace re by e r . 

We have now to examine whether <j>(z) can be properly 
chosen. It is natural to take 

<f>(z)= Z -H/W Mm Um m - ■ ■ ■ (9-8 1 ) 
By its definition it must satisfy (i.). For (ii.) to be true, 
we must have 

E>Me + N>/o, 
which is the trivial condition that there must be enough 
energy to provide each system with the least amount Of 
energy it is permitted to have. Condition (iv.) does not 
appear at first sight inevitable, but must follow from Bohr's 
Correspondence Principle *, for the convergence of the series 
f(z) and g(z) depends on their later terms — that is, those of 
* Bohr, loc. cit. 



tlie Partition of Energy. 471 

large quantum numbers. Condition (v.) is satisfied if not 
all the e's and v's have a common factor. There remains 
(ill/), and here there are trivial analytical difficulties when, 
as in general, M/E and N/E are fractional. 

It is, however, easy to generalize § 6 by replacing [</>(c )] E 
h J 

and letting E, M, N all tend to infinity independently. 
Condition (iii.) is then satisfied, as can be seen by multiplying 
out, and so all the conditions are satisfied, and the final 
results stated above are unaffected. 

Finally, we may observe that all our results can be 
extended at once to an assembly containing any number of 
types of system. If there are M c systems of type C, for 
which the partition function is/ c (3), then 

B.=lW^-log/.(»). 

where 3 is determined by 

E=S M^log/ c ($). 

The formal validity of the proof will require all the 
quantities e c to be commensurable. It will be shown in § 12 
how this restriction may be removed. 

§ 10. Vibrators of two and three degrees of freedom. 

As a first example we take a set of vibrators each of which 
is free to vibrate in a plane under a central force proportional 
to the distance. The sequence of energies is again 0, e, 
2e, ..., but the weights are no longer unity, as the system is 
degenerate. Following the principle laid down in § 2, we 
may evaluate the weights by treating the system as non-de- 
generate and counting the number of different motions 
which have the same total quantum number. Now we can 
quantize the plane vibrators in directions x and y, and as an 
example for the case 2e, we have three alternatives (2e, 0), 
(e, e), (0, 2e). This is easily generalized, and gives to re 
the weight r-f-1. The partition function for these vibrators 
is thus 

/(c) = l + 2* + 3; 2 +4s 3 +..., 



472 Messrs. C. G. Darwin and R. H. Fowler 



on 



From the general theorem (9*7) we at once have 

so that such vibrators have just twice as much energy as 
the line vibrators. 

In exactly the same way we can treat the case of three 
dimensions. To illustrate the weights we again take the 
case of 2e and quantize the svstem in x, y, z. There are 
six alternatives (2e, 0, 0), (0/2e, 0), (0, 6, 2e), (e, €, 0), 
(e, 0, e), (0, e, e). The general form for re is J(r+ l)'(f + 2). 
The partition function is now 

= (l-e)" 3 5 
which leads at once to the expected result 

§ 11. Rotating Molecules. 

Another interesting example to which the calculations at 
once apply is that part of the specific heat of a gas due to 
the rotations of the molecules. Various writers * have 
quantized the motions of a rigid body, and it is found that 
the system has at most two instead of three periods, so that 
it is partly degenerate. We may consider for simplicity a 
diatomic molecule. Then, on account of the small moment 
of inertia about the line of centres, the third degree of 
freedom may be omitted altogether — its quantum of energy 
is too large. A simple calculation then leads to energies of 
rotation e r given by 

^=*5r 2 > c*d 

where I is the moment of inertia about a transverse axis, 
which we shall assume to be independent of r. This is a 
degenerate system, and considerations of the number of cases 
which occur if it is quantized for the two degrees shows 
that the weight to be attached is 2r-f 1. This is on the 
principles suggested by Bohr f with a simplifying modifica- 
tion; for Bohr had to suppose that certain quantized motions 
were excluded for other reasons which are not operative 

* Among others, Ehrenfest, Verh. Deutsch. Phys. Ges. xv. p. 451 
(1913). Epstein, Phys. Zeitschr. xx. p. 289 (1919;. F. Reiche, Ann. 
der Physik, liv. p. 421 (1917). 

f Loc. cit. p. 26. 



the Partition of Energy. 473 

here *. There can, we think, be no question as to the correct- 
ness of the weight 2r+l, bat most recent writers have used 
a factor?*; our formula for the specific heat has therefore 
a rather different value. 

We may now apply our general formulae to this case with 

p r =2r+l, e=rh, 6=^. . . (11-2) 

Then /(^) = l + 3S 6 + 5^ fc + 7S 9fc '+..., . . (H-3) 

E A = M$J^log/(a) (11-31) 

The contribution of the rotations to the molecular specific 
heat, C rot , is dEJdT, where M must be taken as the number 
of molecules in one gramme-molecule of gas. Thus, using 
(7*1), we have 

M.k /. d 2 



5 (^)log/W, • • (H-4) 



Crot (log&J s 
and M£ = R, the usual gas constant. If we write 

*-k5et' (11 ' 5) 



Crot^R^ — logil + Ze-' + Oe-^+le- 9 "-)-...). (11-51) 



then 

da* 

Equation (11*5) shows that, when T— >co, a— >0. It can 
be shown by the application of standard theorems on series f 
that when <r->0, 

C, *->B, (11-52) 

which is the correct limiting value as required by classical 
dynamics. 

In the general case of any body we have three degrees 
of rotational freedom, the motion is simply degenerate J, 
and the energy enters as a sum of square numbers multi- 
plying two units of energy. The motion of the axis of 
symmetry and the motion about the axis of symmetry are 
not independent, and it is impossible therefore for the parti- 
tion function to split up into the product of two partition 
functions which represent the separate contributions of the 
two motions. The result is a double series of the same 
general type as (11*3). 

* Assuming that no extraneous considerations rule out any of these 
states. 

t JBromwich, Infinite Series, p. 132. The theorem is due to Cesaro. 
X Epstein, Phys. Zeit xx. p. 289 (lyl9). 



474 Messrs. C. G. Darwin and R. H. Fowler 



on 



It does not appear profitable to examine these expressions 
further here, since the agreement with experiment is not 
very good at all temperatures. It is to be presumed that 
the assumption of constant moments of inertia is at fault, 
and this is supported by some of the evidence from band 
spectra ; further, it is probable that the case of no rotation 
must be excluded, involving the omission of the first term 
in the partition functions. The discussion of the practical 
applications of these formulae cannot be entered into here. 

§ 12. Assemblies containing free molecules. 

The problems we have so far discussed have all possessed 
the distinguishing characteristic that the temperature is the 
only independent variable. As soon as we treat of free 
molecules this is no longer the case, for now the volume 
must be another independent variable. Nevertheless, as we 
shall see, the same methods of calculation are available. 
The partition is no longer represented exactly by integrals, 
as it was for the quantized motions, but from the nature of 
the case some form of limiting process is required. The 
free molecules cannot of course be regarded as the limit of 
three-dimensional vibrators of low frequency, for they have 
no potential energy to share in the partition. We must 
proceed by the method common to most discussions of the 
distribution laws of classical assemblies — divide up into cells 
the six-dimensional space in which the state of any molecule 
is represented, associate with each cell a certain constant 
value of the energy, and in the limit make all the dimensions 
of all the cells tend to zero *. 

We take an assembly composed of M systems of the type 
A of § 9 and P free-moving monatomic molecules of mass m 
and of small size, the whole enclosed in a vessel of volume V. 
The energy of the molecules is solely their energy of trans- 
lation ; they are supposed to obey the laws of classical 
mechanics (except during their collisions with the A's). In 
order to specify the state of the assembly, we*fcake a six- 
dimensional space of co-ordinates q u q 2 ,...p 3 , the three 
rectangular co-ordinates and momenta of a molecule in the 
vessel. We divide up this space into small cells, 1, 2, 3, ..., 
t ..., of extensions (dq l ...dp z ) u which may or may not be 

* That the limit of the distribution laws worked out for the cells is 
the true distribution law for the actual assembly is an assumption 
implicit in all such discussions. 



the Partition of Energy. 475 

equal. Then by the principles of § 2 the weight factor for 
the tth cell is 

8t= («** •» <fo\ .... ( i2-l) 

provided of course that the cell is relevant to our assembly. 
Only those cells have a weight for which the q } s lie in the 
vessel; but the //s may range over all values from -co to 
+ co, for the method of summation will automatically 
exclude values which could not be allowed. Associated 
with the tth cell there is energy given by 

r ( =^0'i 2 +^ 2 +ps 2 ). . • • (12-iD 

The state of the molecules in the assembly is specified by 

the numbers c l5 c 2 , ... of molecules in cells 1, 2, The 

specification of the A's is as before. The number of weighted 
complexions corresponding to the specification is then 

Mf Pi 

" '— i>o> gl ... > t V^V., . (12-12) 



. a ! fll !../ u ^ •••c 1 U' 2 ! 
where Z r a r = M, $ t c t = F, 2,.a r e r + W* = E - . (12-13) 

In proceeding thus we are constructing an artificial 
assembly in which the energy is taken to have the same 
value f f in all parts of the tth cell, and in which all the f's 
and all the e's can be expressed as multiples of some basal 
unit, without a factor common to thern all. 

This assembly can be made to resemble the real one to 
any standard of approximation required. For such an 
artificial assembly w r e can make use of the whole of our 
machinery. The results all depend on integrals such as 






■*T [««)]*[*«]*, • • a 2 " 2 ) 
where the partition function of the artificial molecules is 

fcO)=2A/<, (1*21) 



and the formulae of § 8 follow at once for E c , (E c — E c ) 2 , c t 
and (<' t — c t ) 2 . These results give completely the exact partition 
laws for any artificial assembly of the type considered. To 
obtain the actual distribution law for the real assembly, we 
must make all the dimensions of all the cells tend to zero, 
and obtain the limit of the partition function. Now, by the 



476 Messrs. G. Gr. Darwin and R. H. Fowler on 

definition of an integral, in the limit A($)-»H($), where 

H(*)= ^e-^W^d* ... d n . (12-3) 

The integration is over the volume V and over all values of 
the p's from -co to + co . This gives at once 

(27nn)3/*V 

In the formulae the functions dh/dS and d 2 h/d§ 2 also occur, 
and it is easily shown directly that their limits are dH/d$ 
and d 2 K/d$ 2 . We may therefore use (12*4) throughout for 
the real assembly; and at once obtain the following ex- 
pressions : 

ET c= g|e/5 =* P * T ' • (12 ' 5) 

_ _ p ^niogjiy * e _ i{iog 1/S) „ ( „ 2+o2+K2) dw dy dz du dv dw ^ 

(12-51) 

= P(^JV^<" 2+ ^>&....^. . . . (12-52) 
The temperature $ is determined by 

E = M4log/(3)+fP— L^, . (12-53) 

and the fluctuation of energy of the molecules in a bath of 
temperature S is given by 

(E -E c )?=a^E c =fPOT . . (12-54) 

If the fluctuation of c^ is evaluated, it takes the simple value 



(e t -Z t y = c ( , (12-55) 

whether it is in a bath or not ; for the second factor, 
analogous to that in (8*4), can be omitted when the cell is 
taken to be of small size. Thus in all cases the possession 
of c t is a normal property of the assembly. These results 
can be readily extended to cases where there is an external 
field of force acting on the molecules. 

By means of this assembly we can establish the meaning 
of 3 in terms of T, by observing that the gas itself constitutes 
a constant volume gas thermometer. It is easy to show that 
the pressure of a gas must be f of the mean kinetic energy 



the Partition of Energy, 477 

in unit volume, that is to say, Jt> = P/ Vlog(l/$). Since the 
gas temperature is measured by the relation pV = P£T, we 
are again led to the relation d = e~ lik . 

We may observe that it is now possible to drop the 
assumption of commensurability, which was necessary in 
the sections which dealt with quantized systems. It was 
there essential, physically speaking, in order that it should 
be possible that the whole of the energy should be held 
somewhere; but as we now have molecules which can hold 
energy in any amounts, it may be dispensed with, the modifi- 
cation being justified on the same assumptions and by 
the same sort of limiting process as have been used in this 
section. Again, we can see that the correct results are 
obtained if H(~) replaces h(z) in (12*2) and all the other 
integrals, even though the interpretation as coefficient in a 
power series is no longer possible, and though the integrand 
is no longer single valued. In such many- valued integrands 
the limiting process shows that we simply require to take 
that value which is real on the positive side of the real axis. 

§ 13. The Maxwell Distribution Law. 

We have carried out the whole process so far with quantized 
systems included in the assembly, but it may be observed 
that it is immediately applicable to an assembly composed 
solely of molecules. If this is done the value of c t in (12*51) 
establishes at once the Maxwell distribution law, and its- 
fluctuation in (12 # 55) proves that it is a normal property of 
the assembly. This is probably the simplest complete proof 
of the ordinary distribution law ; its special advantage is 
that by means of the fluctuations it is easily established that 
the actual distribution will hardly ever be far from the 
average. 

The method can also be made to establish the distribution 
law for a mixture of gases *, and indeed for a mixture of 
any kind, provided that the systems can be considered to 
have separate energies. 

It is also possible to extend the method to cases in which 
the total momentum or angular momentum is conserved, by 
constructing partition functions in more than one independent 
variable. In fact, there will be as many independent 
variables as there are uniform integrals of the dynamical 
equations of the assembly. For simplicity we shall suppose 
that the linear momentum in a given direction is conserved, 

* The effects of the semi-permeable membranes of thermodynamics- 
can be conveniently treated by the partition function. 



478 On the Partition of Energy. 

and let its total amount be Gr. The method now requires 
the averaging process to be applied to expressions depending 
on 



P! 



W 2 , 



where we now have not only 

but also 2, t c t fi t =Q, 

where fi t is the momentum in the given direction of a mole- 
cule in the tth cell. To sum the appropriate expressions we 
must take as our partition function 

With this function C will be the coefficient of z a G in 
[h(z, #)] P , and this can be expressed as a double contour 
integral. So can the other averages, and the usual asymptotic 
expansions can be found. The correct distribution law 
follows on replacing h(z, x) by the integral which is its limit 
when the sizes of the cells tend to zero. This subject lies 
rather outside the theme of the present paper and need not 
be elaborated further. 

§ 16. Summary. 

The whole paper is concerned with a method of calculating 
partitions of energy by replacing the usual calculation, which 
obtains the most probable state, and is mathematically un- 
satisfactory, by a calculation of the average state, which is 
the quantity that is actually required and which can be found 
with, rigour by the use of the multinomial theorem together 
with a certain theorem in complex variable theory. 

After a review of principles and two preliminary examples 
ihe real point of the method is illustrated in § 5. Here 
there are two groups of interacting Planck vibrators of 
different types. It is shown that the partition can be found 
by evaluating the coefficient of a certain power of z in an 
expression which is the product of power series in z. This 
coefficient can be expressed as a contour integral and can be 
evaluated by a well-known method, the " method of steepest 
descents." The result expresses itself naturally in terms of 
a parameter 3 which is identified with temperature measured 
on a scale given by ^~e~ llk . 

The work is extended to cover the partition among more 



The Heterodyne Beat Method. 479 

general quantized systems in §9, and examples are given. 
In § 12 it is shown how it may be made to deal with as- 
semblies composed partly of: free molecules and partly of 
quantized systems. In §13 we deal with extensions possible 
when only molecules are present. 

The methods we have described can also be made to throw 
an interesting light on the statistical foundations of therrao- 
dynamics; but in that connexion many points have arisen 
which require rather careful discussion, and in order not to 
make the present paper too long, we have deferred them to 
a future communication. 

Cambridge, 
May, 1922. 



XLV. The Heterodyne Beat Method and some Applications 
to Physical Measurements. By Maurice H. Belz, M.Sc. 
(Cantab.), Barker Graduate Scholar of the University of 
Sydney *. 

IN a recent paper f, a preliminary account was given of 
the application of the heterodyne beat method to the 
measurement of magnetic susceptibilities. In virtue of the 
importance of the method as a sensitive measure of physical 
quantities, it seems desirable to give a more complete account 
of the principle and of some of the difficulties encountered 
in its application. 

Essentially the method consists of the following arrange- 
ment shown in fig. 1. 

Two oscillating circuits, Set 1 and Set 2, are set up side 
by side and arranged so as to have approximately the same 
frequency. The two sets are loosely coupled so that in the 
telephone included in one of the circuits a resultant beat 
frequency is maintained equal to the difference between the 
frequencies of the fundamentals or overtones in the two 
circuits. If symmetry in the two circuits is essential, direct 
coupling can be replaced by indirect coupling by means of a 
third circuit in which the telephone is placed. In either 
case, when the beat frequency is low enough, an audible note 
will be heard in the telephone, and any changes in the 
constants of either circuit will cause the frequency of 
the audible note to alter by an amount equal to the change 
in frequency of the responsible circuit. This at once provides 

* Communicated by Professor Sir E. Rutherford, F.R.S. 
t Belz, Proc. Camb. Phil. Soc. vol. xxi. part 2 (1922). 



480 Mr. M. H. Belz on the Heterodyne Beat Method 

a very sensitive method. It is now easily possible to maintain 
oscillations of frequencies up to 10 7 per second. Taking the 
case when Set 1 has a frequency of 1,000,000 per second, 
Set 2 a frequency of 1,001,000 per second, the audible note 
will have a frequency of 1,000 per second. If the frequency 
of Set 1 is changed to 1,000,001 per second, the frequency of 
the audible note will now be 999 per second, and this change 
in pitch can readily be observed by comparison with a note 
of standard pitch. 

Fig. 1. 




This sensitive method has been successfully employed by 
Herweg *, Whiddington f , Pungs and Premier J, Falcken- 
burg §, and several others in physical researches, but the 
precautions necessary for steadiness in the beat note have 
never been completely specified. 



Precautions. 

With high frequency oscillations of the order 3 x 10 5 per 
second to 5 X 10 5 per second such as were used in the present 
investigations, electrostatic shielding from all external in- 
fluence was of the first importance. This was ensured by 
placing all the elements of the circuits in earthed metal- 
lined boxes, one of the variable capacities, by means of which 
final small adjustments were made, being provided with a 
long ebonite spindle which projected beyond the containing 
box. With the box closed the note from the telephone T 
was considerably reduced in intensity, and in order to obtain 
the maximum loudness, a small section was removed from 
the box, shielding being maintained by means of a piece of 
fine metal gauze. 

* Herweg, Zeit.f. Phys. vol. iii. p. 36 (1920). 

f Whiddington, Phil. Mag. vol. xi. p. 634 (1920). 

% Pungs and Preuner, Phys. Zeit. vol. xx. p. 543 (1919). 

§ Falckenburg, Ann. d. Phys. vol. lxi. 2, p. 167 (1920). 



and some Applications to Physical Measurements. 481 

Solidity of foundation is a most important requirement. 
In the experiments of Whiddington*, although the apparatus 
was set up on a solid base, vibrations of the building even 
at 2 a.m. proved troublesome. A somewhat similar trouble 
was experienced in some of the earlier experiments when the 
apparatus was installed on the top floor of the laboratory. 
It was found that the vibration of the building caused by 
people walking about the corridors, and by the passage of 
heavy motor traffic, appreciably affected the steadiness of the 
note. Although some of the work was done during the night 
and over the week end, the trouble always persisted. 

Finally the apparatus was transferred to a room on the 
ground floor and supported on stone pillars by means of solid 
rubber pads. The trouble was now completely removed so 
that successful observations could be made during the day 
despite the fact that people were continually walking beside 
the apparatus. 

After lighting the valves, a certain amount of time must 
elapse before the oscillating system has settled down to a 
steady state. This initial variation is due to the heating and 
expansion of the elements of the valves, causing changes in 
the whole capacity linked with the oscillating systems. In 
order to save time, thus sparing the high tension batteries 
and prolonging the life of the valves, the latter were contained 
in small tin boxes, placed outside the large box, which were 
lagged with asbestos and cotton -wool. In this way the heat 
conduction was minimized, and the valves settled down much 
more rapidly. Other conditions being the same, it was found 
that certain valves were less satisfactory than others. For 
some types the settling down process was very long, and by 
the time the valve was set, other things began to vary. 
After long trials with " E," " Fotos," and " A. T. " types, 
it was found that " B " type valves manufactured by the 
General Electric Company gave most satisfactory results, 
settling down most rapidly and remaining steadiest. 

The effects of the changes in the elements of an oscillating 
system on the frequency have been examined by Eccles and 
Vincent f in the case of wave-lengths of 3000 metres. They 
determined that between certain limits for each value of the 
coupling between the plate and grid coils there was a 
particular value of the filament current for which the wave- 
length was a maximum. Working at this value of the 
current it was found possible to hold the beat note steady to 

* Whiddington, loc. cit. 

| Eccles and Vincent, Proc. Roy. Soc. A. vol. xcvi. p. 455 (1919). 

Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2 I 



482 Mr. M. H. Belz on the Heterodyne Beat Method 

one part in 100,000 for several minutes in spite of small 
unavoidable variations. With the frequencies employed in 
the present experiments, however, such a condition could 
not be established. The heating of the valve parts and the 
consequent change in capacity in the system resulting from 
changes in the filament current cause changes in wave-length 
which certainly far outweigh any real change due to increased 
thermionic emission alone. In order then to secure a constant 
filament current, accumulators of 100 ampere-hours capacity 
were employed. These were charged regularly after about 
three days' use, and after the valve had settled down, the 
current from them showed no variation during a single run. 

Faulty contacts of wires joining the elements of the 
circuits were avoided by soldering, the only sources of 
uncertainty being the sliding resistances in the filament 
circuits. These, however, were good types with bright 
surfaces and stiff springs so that the chance of error due to 
change of contact was small. 

The principal cause of variation in the frequencies of the 
circuits was found to be due to variations in the high tension 
batteries. This trouble has been mentioned by Eccles and 
Vincent *. In the present work the plate voltage was 
obtained from trays of portable accumulators of fairly low 
capacity, each tray providing 40 volts. After the valves 
had been burning for an hour or so, taking a current of 
about 10 milliamperes, this voltage began to vary and the 
beat note consequently drifted. However, giving the valves 
time to settle down, a matter of 15 to 20 minutes, it was 
found possible to hold the heterodyne note quite steady for 
intervals of 30 to 60 seconds, and this is ample time in which 
to make a single observation. After about 90 minutes 
burning the variation was too rapid and the batteries had to 
be recharged. The size and consequent capacity of the cells 
of these batteries is limited by the fact that they have to be 
contained in a metal box, and thus this source of variation 
can only be provided for in special cases. 

Technique. 

It is essential to maintain the oscillations generated in the 
circuits at frequencies considerably different from the natural 
frequencies of the coils alone f, that is to say with an 
appreciable capacity in the system, and under these conditions 
the frequency, n, of the oscillations in such a circuit containing 

* Eccles and Vincent, loc. cit. 

+ Cf. Townsend, Phil. Mag. vol. xlii. August (1921). 



and some Applications to Physical Measurements. 483 
inductance L and capacity C, is given very approximately by 

71 = 1/(2^01). 

Changes in n can thus be brought about by changes in 
( ! or L. In the experiments to be described below, the 
changes in n were brought about by variations in L, and in 
this case, with C constant, a small variation, dh, in the 
inductance produces a corresponding change, tin, in the 
frequency given bv 

dn/n=--idL/L (i.) 

The experimental part thus reduces itself to a determination 
of dn. This is accomplished by obtaining beats between the 
heterodyne note and a note o£ constant pitch, and then 
counting the change in the number of beats per second 
caused by the change in inductance. A considerable amount 
of practice in listening is required in order readily to be able 
to adjust the heterodyne note to the pitch of the constant 
note. This note can be very conveniently obtained by means 
of a third set, some distance away from the other sets, 
oscillating with audible frequency, in the plate circuit of 
which a telephone is placed. The intensity of the note 
heard can be altered by adjusting the filament current, and 
in this respect the note is very much more convenient than 
that obtained from a tuning-fork. For it was found that 
the heterodyne note could be more easily brought to tune 
with the standard note, and false beats more readily recognized 
when this latter could be altered so that both notes had 
approximately the same intensity. 

In some of the experiments * it was found impossible to 
obtain a beat note of convenient audible frequency when the 
fundamental frequencies of the oscillations were approxi- 
mately the same. It was observed that, as the capacity of 
one of the sets was altered, only very shrill notes could he 
heard on either side of the very large region of silence. 
This synchronization effect appears to depend on several 
factors, but chiefly on the coupling between the circuits. 
On account of the limited size of the box containing the 
coils, the coupling could not be reduced beyond a certain 
lower limit, and reducing the strengths of the oscillatory 
currents merely reduced the intensity of the limited note. 
In all these cases it was possible to obtain the heterodyne 
note between the fundamental of Set 2 and the first overtone 
of Set 1, which was quite steady and possessed the normal 
region of silence. Since the changes in n were produced by 

* Those in which the determinations of the magnetic susceptibilities 
of certain salts were made, see below. 

2 I 2 



484 Mi\ M. H. Belz on the Heterodyne Beat Method 

variations in the inductance o£ Set 1, this arrangement 
increases the sensitiveness of the method. For let "N" be the 
frequency, determined at the centre of the region of silence, 
of the fundamental oscillation in Set 2, n the frequency of 
the fundamental oscillation in Set 1, then since the rirst 
overtone of Set 1 is employed to produce the note, ~N = 2n« 
Let the frequency of the audible note from the third circuit 
be m. Then when the heterodyne note is adjusted, by 
slightly varying the capacity of Set 2, so that q beats per 
second are counted, the frequency of Set 2 is (N±??2±g) 
— (2n + m±q). If now the frequency of the fundamental 
oscillation of Set 1 is altered by dn per second, the frequency 
of the first overtone is altered by an amount 2dn per 
second, so that the frequency of the heterodyne note is 
now (2n±m + ^)~ (2n±2dn) — (m + q±2dn), whence if a 
change of p beats per second is observed when the induc- 
tance change is accomplished, p = 2dn. The sensitiveness 
is thus doubled, and could similarly be increased by em- 
ploying higher overtones of Set 1. Against this, however, 
is the fact that the notes so obtained are very feeble, and 
counting becomes increasingly difficult. 

From equation (i.) we see that the sensitiveness depends 
on n. It is now possible to maintain oscillations of frequencies 
up to 10 7 per second, but in cases where the change in in- 
ductance is caused by inserting a specimen within the coil, 
there is an upper limit to n determined by the form and 
function of the coil L. It is necessary to divide this coil 
into two parts between which there is no mutual inductance, 
one part L t being coupled to the grid circuit in order to 
maintain the oscillations, the other part L 2 serving as the 
coil in which the inductance changes occur. This latter 
part must be a fairly long coil in order that there may be an 
appreciable region within it through which the magnetic 
field is constant, in which region the specimen is placed. 
On account of the dimensions of this coil, the first part has 
to possess a fairly large inductance in order to get sufficient 
mutual inductance with the grid coil : further, a certain 
amount of coupling is required with Set 2 to produce the 
heterodyne note. 

Experimental. 

In the present experimental arrangements the details of 
the coils are as follows :— 

Coil Lj. 

The coil was 10 cm. long, and consisted of 100 turns of 
22 s.w.g., double cotton covered. It was 



and some Applications to Physical Measurements. 485 

wound on a short length of glass tubing and had an effective 
diameter of 2'10 cm. The self-inductance, employing the 
exact formula of Nagaoka *, viz., 

L s = 47r 2 <2 2 >i 1 2 b. K, 

where L s is the self-inductance of a current sheet of the 
same dimensions as the coil, n 1 the number of turns per cm., 
a the effective radius, b the total length, and K a factor 
depending on the ratio of the diameter of the coil to the 
length, to which was applied the correction for spacing, was 
calculated to be 39,160 cm. The small frequency correction 
was neglected. 

Coil L 2 . 

The coil was 36*70 cm. long, and consisted of 541 turns 
of copper wire, no. 24 s.w.g., silk covered. It was wound 
on a long glass tube, of external diameter 1*00 cm., and 
separated therefrom by means of a layer of paraffined paper. 
The effective diameter (2a) of the coil was 1*105 cm., and 
self-inductance, calculated as above, was 92,430 cm. 

The total inductance L ( = L 1 + L 2 ) is thus 131,600 cm. 

The coil L 2 was outside the box containing the rest of 
the circuits, and was shielded from external electrostatic 
influences by means of an enveloping earthed metal cylinder. 

Coil L 3 . 

The length was 9 cm., and the coil consisted of 90 turns 
of copper wire. no. 22 s.w.g., double cotton covered. It was 
wound on a short, length of glass tubing and had an effective 
diameter 4*13 cm. The self-inductance was similarly cal- 
culated to be 124,600 cm. 

The capacities employed had a range of 100 to 1200 
microfarads and were provided with a slow movement. 

Changes in the frequency of the oscillations of Set 1 
brought about by the insertion of a specimen within the 
coil L 2 may be due to three causes : — 

(a) In the first place, if the coil is not shielded from the 
electrostatic effect of the specimen, the self capacity of 
the coil will be changed. In order to observe changes 
of inductance alone, it is necessary to guard against this 
possibility. This was done by depositing a thin layer of 
platinum f on the outside of the glass tube on which the 
coil L was wound, and earthing. The thickness of the deposit, 
obtained by weighing, was 7 x 10 ~ 6 cm. It is necessary to 

* Nagaoka, Jour. Coll. Sci. Tokyo, xxvii. art. 6, p. 18 (1909). 
t The function of the paraffined paper was to prevent any possible 
short-circuiting 1 of the coil through the layer of platinum. 



486 Mr. M. H. Belz on the Heterodyne Beat Method 

determine the effect of this shield on the strength of the 
magnetic field within. The magnetic force, H t , at a depth t 
in a mass of metal is related to the force, H , at the surface 
by the equation 



H, = IV 



V -ST-' 



in which //,, cr represent the permeability and specific 
resistance respectively of the metal, and p = 27rn, n being 
the frequency. Taking w = 4'84 x 10° per second, the largest 
frequency used, and